SUBALGEBRAS OF C^{∗}-ALGEBRAS III:

MULTIVARIABLE OPERATOR THEORY

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

18 May 1997

Abstract. A d-contraction is a d-tuple (T_{1}, …, T_{d}) of mutually commuting operators acting on a common Hilbert space H such that

∥T_{1}ξ_{1} + T_{2}ξ_{2} + ⋯ + T_{d}ξ_{d}∥^{2} ≤ ∥ξ_{1}∥^{2} + ∥ξ_{2}∥^{2} + ⋯ + ∥ξ_{d}∥^{2}

for all ξ_{1}, ξ_{2}, …, ξ_{d} ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball B_{d} in complex d-space, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the d-shift, a distinguished d-contraction which acts on a new H^{2} space associated with B_{d}, and which is the higher dimensional counterpart of the unilateral shift. H^{2} and the d-shift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the d-shift relative to its generated C^{∗}-algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.

Contents

Introduction Part I: Function theory

1. Basic properties of H^{2}

2. Multipliers and the d-dimensional shift

3. von Neumann’s inequality and the sup norm

4. Maximality of the H^{2} norm Part II: Operator theory

5. The Toeplitz C^{∗}-algebra

6. d-contractions and A-morphisms

7. The d-shift as an operator space

8. Various applications Appendix: Trace estimates References

This research was supported by NSF grant DMS-9500291

Typeset by AMS-TEX

Introduction.

This paper concerns function theory and operator theory relative to the unit ball in complex d-space ℂ^{d}, d = 1, 2, . . .. A d-contraction is a d-tuple (T_{1}, …, T_{d}) of mutually commuting operators acting on a common Hilbert space H satisfying

∥T_{1}ξ_{1} + ⋯ + T_{d}ξ_{d}∥^{2} ≤ ∥ξ_{1}∥^{2} + ⋯ + ∥ξ_{d}∥^{2},

for every ξ_{1}, …, ξ_{d} ∈ H. This inequality simply means that the “row operator” defined by the d-tuple, viewed as an operator from the direct sum of d copies of H to H, is a contraction. It is essential that the component operators commute with one another.

We show that there exist d-contractions which are not polynomially bounded in the sense that there is no constant K satisfying

∥f(T_{1}, …, T_{d})∥ ≤ K sup{|f(z_{1}, …, z_{d})| : |z_{1}|^{2} + ⋯ + |z_{d}|^{2} ≤ 1}

for every polynomial f. In fact, we single out a particular d-contraction (S_{1}, …, S_{d}) (called the d-shift) which is not polynomially bounded but which gives rise to the appropriate version of von Neumann’s inequality with constant 1: for every dcontraction (T_{1}, …, T_{d}) one has

∥f(T_{1}, …, T_{d})∥ ≤ ∥f(S_{1}, …, S_{d})∥

for every polynomial f. Indeed the indicated homomorphism of commutative operator algebras is completely contractive.

The d-shift acts naturally on a space of holomorphic functions defined on the open unit ball B_{d} ⊆ ℂ^{d}, which we call H^{2}. This space is a natural generalization of the familiar Hardy space of the unit disk, but it differs from other “H^{2}” spaces in several ways. For example, unlike the space H^{2}(∂B_{d}) associated with normalized surface area on the sphere or the space H^{2}(B_{d}) associated with volume measure over the interior, H^{2} is not associated with any measure on ℂ^{d}. Consequently, the associated multiplication operators (the component operators of the d-shift) do not form a subnormal d-tuple. Indeed, since the naive form of von Neumann’s inequality described above fails, no effective model theory in dimension d ≥ 2 could be based on subnormal operators. Thus by giving up the requirement of subnormality for models, one gains a theory in which models not only exist in all dimensions but are unique as well.

In the first part of this paper we work out the basic theory of H^{2} and its associated multiplier algebra, and we show that the H^{2} norm is the largest Hilbert norm on the space of polynomials which is appropriate for the operator theory of d-contractions.

In part II we emphasize the role of “A-morphisms”. These are completely positive linear maps of the d-dimensional counterpart of the Toeplitz C^{∗}-algebra which bear a particular relation to the d-shift. Every d-contraction corresponds to a unique A-morphism, and on that observation we base a model theory for d-contractions which provides an appropriate generalization of the Sz.-Nagy Foias theory of contractions [43] to arbitrary dimension d ≥ 1 (see section 8). In section 7 we introduce a sequence of numerical invariants E_{n}(S), n = 1, 2, … for arbitrary operator spaces S. We show that the d-dimensional operator space S_{d} generated by the d-shift is maximal in the sense that E_{n}(S_{d}) ≥ E_{n}(S) for every n ≥ 1 and for every ddimensional operator space S consisting of mutually commuting operators. More significantly, we show that when d ≥ 2, S_{d} is characterized by this maximality property. That characterization fails for single operators (i.e. one-dimensional operator spaces). We may conclude that, perhaps contrary to one’s function-theoretic intuition, there is more uniqueness in dimension d ≥ 2 than there is in dimension one.

Since this paper is a logical sequel to [3], [4] and so many years have passed since the publication of its two predecessors, it seems appropriate to comment on its relationship to them. On the one hand, we have come to the opinion that the program proposed in [4, chapter 1] for carrying out dilation theory in higher dimensions must be modified. That program gives necessary and sufficient conditions for finding normal dilations in multivariable operator theory. However, the results below provide two reasons why normal dilations are inappropriate for commutative sets of operators associated with the unit ball B_{d}. First, they may not exist (a d-contraction need not have a normal dilation with spectrum in ∂B_{d}, c.f. Remark 3.13) and second, when they do exist they are not unique (there can be many normal dilations of a given d-contraction which have the stated properties but which are not unitarily equivalent to each other).

On the other hand, the results of this paper also demonstrate that other aspects of the program of [3,4] are well-suited for multivariable operator theory. For example, we will see that boundary representations...the noncommutative counterparts of Choquet boundary points in the commutative theory of function spaces...play an important role in the operator theory of B_{d}. Boundary representations serve to explain the notable fact that in higher dimensions there is more uniqueness than there is in dimension one (cf. Theorem 7.7 and its Corollary), and they provide concrete information about the absence of inner functions for the d-shift (cf. Proposition 8.13).

We were encouraged to return to these problems by recent results in the theory of E_{0}-semigroups. There is a dilation theory by which, starting with a semigroup of completely positive maps of B(H), one obtains an E_{0}-semigroup as its “minimal dilation” [14,6,7,8,9,10]. In its simplest form, this dilation theory starts with a normal completely positive map P : B(H) → B(H) satisfying P (1) = 1 and constructs from it a unique endomorphism of B(K) where K is a Hilbert space containing H. When one looks closely at this procedure one sees that there should be a corresponding dilation theory for sets of operators such as d-contractions.

We have reported on some of these results in a conference at the Fields institute in Waterloo in early 1995. That lecture concerned the dilation theory of semigroups of completely positive maps, A-morphisms and the issue of uniqueness. However, at that time we had not yet reached a definitive formulation of the application to operator theory.

There is a large literature relating to von Neumann’s inequality and dilation theory for sets of operators, and no attempt has been made to compile a comprehensive list of references here. More references can be found in [26],[27]. Finally, I want to thank Raúl Curto for bringing me back up to date on the literature of multivariable operator theory.

1. Basic properties of H^{2}. Throughout this paper we will be concerned with function theory and operator theory as it relates to the unit ball B_{d} in complex d-dimensional space ℂ^{d}, d = 1, 2, …,

B_{d} = {z = (z_{1}, z_{2}, …, z_{d}) ∈ ℂ^{d} : ∥z∥ < 1},

where ∥z∥ denotes the norm associated with the usual inner product in ℂ^{d}

∥z∥^{2} = |z_{1}|^{2} + |z_{2}|^{2} + ⋯ + |z_{d}|^{2}.

In dimension d = 1 there is a familiar Hardy space which can be defined in several ways. We begin by reiterating one of the definitions of H^{2} in a form that we will generalize verbatim to higher dimensions. Let P be the algebra of all holomorphic polynomials f in a single complex variable z. Every f ∈ P has a finite Taylor series expansion

f(z) = a_{0} + a_{1}z + ⋯ + a_{n}z^{n}

and we may define the norm ∥f∥ of such a polynomial as the ℓ^{2} norm of its sequence of Taylor coefficients

(1.1)  ∥f∥^{2} = |a_{0}|^{2} + |a_{1}|^{2} + ⋯ + |a_{n}|^{2}.

The norm ∥f∥ is of course associated with an inner product on P, and the completion of P in this norm is the Hardy space H^{2}. It is well known that the elements of H^{2} can be realized concretely as analytic functions

f : {|z| < 1} → ℂ

which obey certain growth conditions near the boundary of the unit disk.

Now consider the case of dimension d > 1. P will denote the algebra of all complex holomorphic polynomials f in the variable z = (z_{1}, z_{2}, …, z_{d}). Every such polynomial f has a unique expansion into a finite series

(1.2)  f(z) = f_{0}(z) + f_{1}(z) + ⋯ + f_{n}(z)

where f_{k} is a homogeneous polynomial of degree k. We refer to (1.2) as the Taylor series of f.

Definition 1.3. Let V be a complex vector space. By a Hilbert seminorm on V we mean a seminorm which derives from a positive semidefinite inner product ⟨·, ·⟩ on V by way of

∥x∥ = ⟨x, x⟩^{1/2} , x ∈ V.

We will define a Hilbert seminorm on P by imitating formula (1.1), where c_{k} is replaced with f_{k}. To make that precise we must view the expansion (1.2) in a somewhat more formal way. The space E = ℂ^{d} is a d-dimensional vector space having a distinguished inner product

⟨z, w⟩ = z_{1}w̄_{1} + z_{2}w̄_{2} + ⋯ + z_{d}w̄_{d}.

For each n = 1, 2, … we write E^{n} for the symmetric tensor product of n copies of E. E^{0} is defined as the one dimensional vector space ℂ with its usual inner product. For n ≥ 2, E^{n} is the subspace of the full tensor product E^{⊗n} consisting of all vectors fixed under the natural representation of the permutation group S_{n},

E^{n} = {ξ ∈ E^{⊗n} : U_{π}ξ = ξ, π ∈ S_{n}},

U_{π} denoting the isomorphism of E^{⊗n} defined on elementary tensors by

U_{π}(z_{1} ⊗ z_{2} ⊗ ⋯ ⊗ z_{n}) = z_{π^{−1}(1)} ⊗ z_{π^{−1}(2)} ⊗ ⋯ ⊗ z_{π^{−1}(n)}, z_{1} ∈ E.

For a fixed vector z ∈ E we will use the notation

z^{n} = z^{⊗n} ∈ E^{n}

for the n-fold tensor product of copies of z (z^{0} ∈ E^{0} is defined as the complex number 1). E^{n} is linearly spanned by the set {z^{n} : z ∈ E}, n = 0, 1, 2, . . ..

Now every homogeneous polynomial g : E → ℂ of degree k determines a unique linear functional g̃ on E^{k} by

g(z) = g̃(z^{k}), z ∈ E

(the uniqueness of g̃_{k} follows from the fact that E^{k} is spanned by {z^{k} : z ∈ E}), and thus the Taylor series (1.2) can be written in the form

f(z) = ∑_{k=0}^{n} f̃_{k}(z^{k}), z ∈ E,

where f̃_{k} is a uniquely determined linear functional on E^{k} for each k = 0, 1, …, n. Finally, if we bring in the inner product on E then E (resp. E^{⊗k}) becomes a d-dimensional (resp. d^{k}-dimensional) complex Hilbert space. Thus the subspace E^{k} ⊆ E^{⊗k} is also a finite dimensional Hilbert space in a natural way. Making use of the Riesz lemma, we find that there is a unique vector ξ_{k} ∈ E^{k} such that

f̃_{k}(z^{k}) = ⟨z^{k}, ξ_{k}⟩ , z ∈ E,

and finally the Taylor series for f takes the form

(1.4)  f(z) = ∑_{k=0}^{n} ⟨z^{k}, ξ_{k}⟩ ,  z ∈ E.

We define a Hilbert seminorm on P as follows

(1.5)  ∥f∥^{2} = ∥ξ_{0}∥^{2} + ∥ξ_{1}∥^{2} + ⋯ + ∥ξ_{n}∥^{2}.

The seminorm ∥ · ∥ is obviously a norm on P in that ∥f∥ = 0 ⟹ f = 0.

Definition 1.6. H_{d}^{2} is defined as the Hilbert space obtained by completing P in the norm (1.5).

When there is no possibility of confusion concerning the dimension we will abbreviate H_{d}^{2} with the simpler H^{2}. We first point out that the elements of H^{2} can be identified with the elements of the symmetric Fock space over E,

F_{+}(E) = E^{0} ⊕ E^{1} ⊕ E^{2} ⊕ …

the sum on the right denoting the infinite direct sum of Hilbert spaces.

Proposition 1.7. For every f ∈ P let Jf be the element of F_{+}(E) defined by

Jf = (ξ_{0}, ξ_{1}, …)

where ξ_{0}, ξ_{1}, … is the sequence of Taylor coefficients defined in (1.4), continued so that ξ_{k} = 0 for k > n. Then J extends uniquely to an anti-unitary operator mapping H^{2} onto F_{+}(E).

proof. The argument is perfectly straightforward, once one realizes that J is not linear but anti-linear □

We can also identify the elements of H^{2} in more concrete terms as analytic functions defined on the ball B_{d}:

Proposition 1.8. Every element of H^{2} can be realized as an analytic function in B_{d} having a power series expansion of the form

f(z) = ∑_{k=0}^{∞} ⟨z^{k}, ξ_{k}⟩ z = (z_{1}, …, z_{d}) ∈ B_{d}

where the H^{2} norm of f is given by ∥f∥^{2} = ∑_{k} ∥ξ_{k}∥^{2} < ∞. Such functions f satisfy a growth condition of the form

|f(z)| ≤ \frac{∥f∥}{\sqrt{1 − ∥z∥^{2}}} , z ∈ B_{d}.

proof. Because of Proposition (1.7) the elements of H^{2} can be identified with the formal power series having the form

(1.9)  f(z) = ∑_{k=0}^{∞} ⟨z^{k}, ξ_{k}⟩ ,

where the sequence ξ_{k} ∈ E^{k} satisfies

(1.10)  ∑_{k=0}^{∞} ∥ξ_{k}∥^{2} = ∥f∥^{2} < ∞.

Because of (1.10) the series in (1.9) is easily seen to converge in B_{d} and satisfies the stated growth condition.

In more detail, since the norm of a vector in E^{k} of the form z^{k}, z ∈ E, satisfies

∥z^{k}∥^{2} = ⟨z^{k}, z^{k}⟩ = ⟨z, z⟩^{k} = ∥z∥^{2k},

we find that

| ⟨z^{k}, ξ_{k}⟩ | ≤ ∥z^{k}∥ · ∥ξ_{k}∥ ≤ ∥z∥^{k}∥ξ_{k}∥,

and hence for all z ∈ E satisfying ∥z∥ < 1 we have

∑_{k=0}^{∞} | ⟨z^{k}, ξ_{k}⟩ | ≤ (∑_{k=0}^{∞} ∥z∥^{2k})^{1/2}(∑_{k=0}^{∞} ∥ξ_{k}∥^{2})^{1/2} = (1 − ∥z∥^{2})^{−1/2}∥f∥,

as asserted □

We will make frequent use of the following family of functions in H^{2}. For every x ∈ B_{d} define u_{x} : B_{d} → ℂ by

(1.11) u_{x}(z) = (1 − ⟨z, x⟩)^{−1}, ∥z∥ < 1.

u_{x}(z) is clearly analytic in z and co-analytic in x. The useful properties of the set of functions {u_{x} : x ∈ B_{d}} are summarized in the following proposition, which gives the precise sense in which H^{2} is characterized in abstract terms by the positive definite reproducing kernel k : B_{d} × B_{d} → ℂ,

k(x, y) = (1 − ⟨x, y⟩)^{−1}.

Proposition 1.12. u_{x} belongs to H^{2} for every x ∈ B_{d} and these functions satisfy

(1.13)  ⟨u_{x}, u_{y}⟩ = (1 − ⟨y, x⟩)^{−1}.

H^{2} is spanned by {u_{x} : x ∈ B_{d}}, and for every f ∈ H^{2} we have

(1.14) f(z) = ⟨f, u_{z}⟩ , z ∈ B_{d}.

Moreover, if K is any Hilbert space spanned by a subset of its elements {v_{x} : x ∈ B_{d}} which satisfy

⟨v_{x}, v_{y}⟩ = (1 − ⟨y, x⟩)^{−1}, x, y ∈ B_{d},

then there is a unique unitary operator W : H^{2} → K such that W u_{x} = v_{x}, x ∈ B_{d}.

proof. The proof is straightforward. For example, to see that u_{x} belongs to H^{2} we simply examine its Taylor series

u_{x}(z) = (1 − ⟨z, x⟩)^{−1} = ∑_{k=0}^{∞} ⟨z, x⟩^{k} .

Noting that ⟨z, x⟩^{k} = ⟨z^{k}, x^{k}⟩_{E^{k}} we can write

u_{x}(z) = ∑_{k=0}^{∞} ⟨z^{k}, x^{k}⟩_{E^{k}} .

This shows that the sequence of Taylor coefficients of u_{x} is

Ju_{x} = (1, x, x^{2}, …) ∈ F_{+}(E).

Hence u_{x} belongs to H^{2} and we have

⟨u_{x}, u_{y}⟩ = ⟨Ju_{y}, Ju_{x}⟩_{F_{+}(E)} = ∑_{k=0}^{∞} ⟨y, x⟩^{k} = (1 − ⟨y, x⟩)^{−1}.

Formula (1.13) follows.

Similarly, a direct application of Proposition 1.7 establishes (1.14). From the latter it follows that {u_{x} : x ∈ B_{d}} spans H^{2}. Indeed, if f is any function in H^{2} which is orthogonal to every u_{x} then

f(z) = ⟨f, u_{z}⟩ = 0 for every z ∈ B_{d},

hence f = 0.

Finally, the second paragraph is obvious from the fact that for every finite subset x_{1}, …, x_{n} ∈ B_{d} and c_{1}, …, c_{n} ∈ ℂ we have

∥c_{1}u_{x_{1}} + ⋯ + c_{n}u_{x_{n}}∥^{2} = ∥c_{1}v_{x_{1}} + ⋯ + c_{n}v_{x_{n}}∥^{2},

which is apparent after expanding both sides and comparing inner products □

The H^{2} norm is invariant under the natural action of the unitary group of ℂ^{d}, as summarized by Corollary. Let V be a unitary operator on the Hilbert space E = ℂ^{d}. Then there is a unique unitary operator Γ(V) ∈ B(H^{2}) satisfying

(1.15)  Γ(V)u_{x} = u_{V x},  x ∈ B_{d}.

Γ is a strongly continuous unitary representation of U(ℂ^{d}) on H^{2} whose action on functions is given by

(1.16)  Γ(V)f(z) = f(V^{−1}z),  z ∈ B_{d}, f ∈ H^{2}.

proof. Fix V ∈ U(ℂ^{d}). For any x, y ∈ B_{d} we have

⟨u_{V x}, u_{V y}⟩ = (1 − ⟨V y, V x⟩)^{−1} = (1 − ⟨y, x⟩)^{−1} = ⟨u_{x}, u_{y}⟩ .

It follows from Proposition 1.12 that there is a unique unitary operator Γ(V) ∈ B(H^{2}) satisfying (1.15). It is clear from (1.15) that Γ(V_{1}V_{2}) = Γ(V_{1})Γ(V_{2}), and strong continuity follows from the fact that

⟨Γ(V)u_{x}, u_{y}⟩ = ⟨u_{V x}, u_{y}⟩ = (1 − ⟨y, V x⟩)^{−1}

is continuous in V for fixed x, y ∈ B_{d}, together with the fact that H^{2} is spanned by {u_{z} : z ∈ B_{d}}.

Finally, from (1.14) we see that for every f ∈ H^{2} and every z ∈ B_{d},

f(V^{−1}z) = ⟨f, u_{V^{−1}z}⟩ = ⟨f, Γ(V^{−1})u_{z}⟩ = ⟨f, Γ(V)^{∗}u_{z}⟩
= ⟨Γ(V)f, u_{z}⟩ = (Γ(V)f)(z),

proving (1.16) □

2. Multipliers and the d-dimensional shift.

By a multiplier of H^{2} we mean a complex-valued function f : B_{d} → ℂ with the property

f · H^{2} ⊆ H^{2}.

The set of multipliers is a complex algebra of functions defined on the ball B_{d} which contains the constant functions, and since H^{2} itself contains the constant function 1 it follows every multiplier must belong to H^{2}. In particular, multipliers are analytic functions on B_{d}.

Definition 2.1. The algebra of all multipliers is denoted M. H^{∞} will denote the Banach algebra of all bounded analytic functions f : B_{d} → ℂ with norm

∥f∥_{∞} = sup_{∥z∥<1} |f(z)|.

The following result implies that M ⊆ H^{∞} and the inclusion map of M in H^{∞} becomes a contraction after one endows M with its natural norm.

Proposition 2.2. Every f ∈ M defines a unique bounded operator M_{f} on H^{2} by way of

M_{f} : g ∈ H^{2} → f · g ∈ H^{2}.

The natural norm in M

∥f∥_{M} = sup{∥f · g∥ : g ∈ H^{2}, ∥g∥ ≤ 1}

satisfies

∥f∥_{M} = ∥M_{f}∥,

the right side denoting the operator norm in B(H^{2}), and we have

∥f∥_{∞} ≤ ∥f∥_{M}, f ∈ M.

proof. Fix f ∈ M. Notice first that if g is an arbitrary function in H^{2} then by (1.16) we have

(2.3)  ⟨M_{f}g, u_{z}⟩ = ⟨f · g, u_{z}⟩ = f(z)g(z).

A straightforward application of the closed graph theorem (which we omit) now shows that the operator M_{f} is bounded.

It is clear that ∥f∥_{M} = ∥M_{f}∥. We claim now that for each x ∈ B_{d} one has

(2.4)  M_{f}^{∗}u_{x} = f̄(x)u_{x}.

Indeed, since H^{2} is spanned by {u_{y} : y ∈ B_{d}} it is enough to show that

⟨M_{f}^{∗}u_{x}, u_{y}⟩ = f̄(x) ⟨u_{x}, u_{y}⟩ , y ∈ B_{d}.

For fixed y the left side is

⟨u_{x}, f · u_{y}⟩ = \overline{⟨f · u_{y}, u_{x}⟩}

By (1.16) the latter is

\overline{f(x)u_{y}(x)} = f̄(x)\overline{(1 − ⟨x, y⟩)^{−1}} = f̄(x)(1 − ⟨y, x⟩)^{−1} = f̄(x) ⟨u_{x}, u_{y}⟩ ,

and (2.4) follows.

Finally, (2.4) implies that for every x ∈ B_{d} we have

|f(x)| = ∥M_{f}^{∗}u_{x}∥/∥u_{x}∥ ≤ ∥M_{f}^{∗}∥ = ∥M_{f}∥ = ∥f∥_{M}

as required □

We turn now to the definition of the d-dimensional analogue of the unilateral shift. Let e_{1}, e_{2}, …, e_{d} be an orthonormal basis for E = ℂ^{d}, and define z_{1}, z_{2}, …, z_{d} ∈ P by

z_{k}(z) = ⟨z, e_{k}⟩ , x ∈ ℂ^{d}.

Such a d-tuple of linear functionals will be called a system of coordinate functions. If z′_{1}, z′_{2}, …, z′_{d} is another system of coordinate functions then there is a unique unitary operator V ∈ B(ℂ^{d}) satisfying

(2.5)  z′_{k}(x) = z_{k}(V^{−1}x),  1 ≤ k ≤ d, x ∈ ℂ^{d}.

Proposition 2.6. Let z_{1}, z_{2}, …, z_{d} be a system of coordinate functions for ℂ^{d}. Then for every complex number a and polynomials f_{1}, f_{2}, …, f_{d} ∈ P we have

∥a · 1 + z_{1}f_{1} + ⋯ + z_{d}f_{d}∥^{2} ≤ |a|^{2} + ∥f_{1}∥^{2} + ⋯ + ∥f_{d}∥^{2},

∥ · ∥ denoting the norm in H^{2}.

proof. We claim first that each z_{k} is a multiplier. Indeed, if f ∈ H^{2} has Taylor series

f(x) = ∑_{n=0}^{∞} ⟨z^{n}, ξ_{n}⟩

with ∑_{n} ∥ξ_{n}∥^{2} = ∥f∥^{2} < ∞ then we have

(2.7)  z_{k}(x)f(x) = ∑_{n=0}^{∞} ⟨x, e_{k}⟩ ⟨x^{n}, ξ_{n}⟩ .

Now

⟨x, e_{k}⟩ ⟨x^{n}, ξ_{n}⟩ = ⟨x^{n+1}, e_{k} ⊗ ξ_{n}⟩ .

So if e_{k} · ξ_{n} denotes the projection of the vector e_{k} ⊗ ξ_{n} ∈ E ⊗ E^{n} to the subspace E^{n+1} then (2.7) becomes

z_{k}(x)f(x) = ∑_{n=0}^{∞} ⟨x^{n+1}, e_{k} · ξ_{n}⟩ .

Since

∑_{n=0}^{∞} ∥e_{k} · ξ_{n}∥^{2} ≤ ∑_{n=0}^{∞} ∥ξ_{n}∥^{2} = ∥f∥^{2}

it follows that z_{k}f ∈ H^{2} and in fact

∥z_{k}f∥ ≤ ∥f∥, f ∈ H^{2}.

Thus each multiplication operator M_{z_{k}} is a contraction in B(H^{2}). Consider the Hilbert space

K = ℂ ⊕ \underbrace{H^{2} ⊕ ⋯ ⊕ H^{2}}_{d times}

and the operator T : K → H^{2} defined by

T (a, f_{1}, …, f_{d}) = a · 1 + z_{1}f_{1} + ⋯ + z_{d}f_{d}.

The assertion of Proposition 2.6 is that ∥T ∥ ≤ 1. In fact, we show that the adjoint of T, T^{∗} : H^{2} → K, is an isometry. A routine computation implies that for all f ∈ H^{2} we have

T^{∗}f = (⟨f, 1⟩ , S_{1}^{∗}f, …, S_{d}^{∗}f) ∈ K,

where we have written S_{k} for the multiplication operator M_{z_{k}}, k = 1, …, d. Hence T T^{∗} ∈ B(H^{2}) is given by

T T^{∗} = E_{0} + S_{1}S_{1}^{∗} + ⋯ + S_{d}S_{d}^{∗}

where E_{0} is the projection on the one-dimensional space of all constant functions in H^{2}. We establish the key assertion as a lemma for future reference.

Lemma 2.8. Let z_{1}, …, z_{d} be a system of coordinate functions for ℂ^{d} and let S_{k} = M_{z_{k}}, k = 1, 2, …, d. Let E_{0} be the projection onto the one-dimensional space of constant functions in H^{2}. Then

E_{0} + S_{1}S_{1}^{∗} + … S_{d}S_{d}^{∗} = 1.

proof. Since H^{2} is spanned by {u_{z} : z ∈ B_{d}} it is enough to show that for all x, y ∈ B_{d} we have

(2.9)  ⟨E_{0}u_{x}, u_{y}⟩ + ∑_{k=1}^{d} ⟨S_{k}S_{k}^{∗}u_{x}, u_{y}⟩ = ⟨u_{x}, u_{y}⟩ .

Since each S_{k} is a multiplication operator, formula (2.4) implies that

S_{k}^{∗}u_{x} = z̄_{k}(x)u_{x} = ⟨e_{k}, x⟩ u_{x},

for x ∈ B_{d}. Thus we can write

∑_{k=1}^{d} ⟨S_{k}S_{k}^{∗}u_{x}, u_{y}⟩ = ∑_{k=1}^{d} ⟨S_{k}^{∗}u_{x}, S_{k}^{∗}u_{y}⟩ = ∑_{k=1}^{d} ⟨e_{k}, x⟩ ⟨y, e_{k}⟩ ⟨u_{x}, u_{y}⟩
= ⟨y, x⟩ ⟨u_{x}, u_{y}⟩ = ⟨y, x⟩ (1 − ⟨y, x⟩)^{−1}.

On the other hand, noting that u_{0} = 1 and ∥u_{0}∥ = 1, the projection E_{0} is given by E_{0}(f) = ⟨f, u_{0}⟩ u_{0}, f ∈ H^{2}. Hence

⟨E_{0}u_{x}, u_{y}⟩ = ⟨u_{x}, u_{0}⟩ ⟨u_{0}, u_{y}⟩ = 1

because ⟨u_{x}, u_{0}⟩ = (1 − ⟨x, 0⟩)^{−1} = 1 for every x ∈ B_{d}. It follows that

⟨E_{0}u_{x}, u_{y}⟩ + ∑_{k=1}^{d} ⟨S_{k}S_{k}^{∗}u_{x}, u_{y}⟩ = 1 + ⟨y, x⟩ (1 − ⟨y, x⟩)^{−1} = (1 − ⟨y, x⟩)^{−1}
= ⟨u_{x}, u_{y}⟩ ,

as asserted □ That completes the proof of Proposition 2.6 □

Definition 2.10. Let z_{1}, …, z_{d} be a system of coordinate functions for ℂ^{d} and let S_{k} = M_{z_{k}}, k = 1, 2, …, d. The d-tuple of operators

S̄ = (S_{1}, S_{2}, …, S_{d})

is called the d-dimensional shift or, briefly, the d-shift.

Remarks. The component operators S_{1}, …, S_{d} of the d-shift are mutually commuting contractions in B(H^{2}) which satisfy

S_{1}S_{1}^{∗} + ⋯ + S_{d}S_{d}^{∗} = 1 − E_{0}

where E_{0} is the projection onto the space of constant functions in H^{2}. In particular, we conclude from Proposition 2.6 that for any f_{1}, …, f_{d} ∈ H^{2}

∥S_{1}f_{1} + ⋯ + S_{d}f_{d}∥^{2} ≤ ∥f_{1}∥^{2} + ⋯ + ∥f_{d}∥^{2}.

Notice too that if we replace z_{1}, …, z_{d} with a different set of coordinate functions z′_{1}, …, z′_{d} for ℂ^{d} then then the operators (S_{1}, …, S_{d}) change to a new d-shift (S′_{1}, …, S′_{d}). However, this change is not significant by virtue of the relation between z_{k} and z′_{k}. More precisely, letting V be the unitary operator defined on ℂ^{d} by (2.5), one finds that

Γ(V)S_{k}Γ(V)^{−1} = S′_{k}, k = 1, 2, …, d,

that is, (S′_{1}, …, S′_{d}) and (S_{1}, …, S_{d}) are unitarily equivalent by way of a natural unitary automorphism of H^{2}. In this sense we may speak of the d-shift acting on H_{d}^{2}. In particular, we may conclude that each component operator S_{i} is unitarily equivalent to every other one S_{j}, 1 ≤ j ≤ d.

Finally, if f is any polynomial in P then we may express M_{f} as a polynomial in the operators S_{1}, …, S_{d} as follows. We find a polynomial function g(w_{1}, …, w_{d}) of d complex variables with the property that f is the composite function of g with the coordinate functions z_{1}, …, z_{d},

f(x) = g(z_{1}(x), …, z_{d}(x)), x ∈ B_{d}.

Once this is done the multiplication operator M_{f} becomes the corresponding polynomial in the operators S_{1}, …, S_{d}:

M_{f} = g(S_{1}, …, S_{d}).

We emphasize that in the higher dimensional cases d ≥ 2, the operator norm ∥M_{f}∥ can be larger than the sup norm ∥f∥_{∞} (see section 3 below). On the other hand, in all dimensions the spectral radius r(M_{f}) of any polynomial multiplication operator satisfies

(2.11)  r(M_{f}) = sup_{z∈B_{d}} |f(z)|.

In the following result we establish the formula (2.11). That follows from a straightforward application of the Gelfand theory of commutative Banach algebras and we merely sketch the details.

Proposition 2.12. Let A be the norm closed subalgebra of B(H^{2}) generated by the multiplication operators M_{f}, f ∈ P.

Every element of A is a multiplication operator M_{f} for some f ∈ M which extends continuously to the closed ball B̄_{d}, and there is a natural homeomorphism of the closed unit ball onto the space σ(A) of all complex homorphisms of A, x ↦ ω_{x}, defined by

ω_{x}(M_{f}) = f(x), ∥x∥ ≤ 1.

For every such f ∈ M one has

lim_{n→∞} ∥M_{f}^{n}∥^{1/n} = sup_{∥x∥<1} |f(x)|.

proof. Since the mapping f ∈ M ↦ M_{f} ∈ B(H^{2}) is an isometric representation of the multiplier algebra on H^{2} which carries the unit of M to that of B(H^{2}), it is enough to work within M itself. That is, we may consider A to be the closure in M of the algebra of polynomials, and basically we need to identify its maximal ideal space.

Because of the inequality ∥f∥_{∞} ≤ ∥f∥_{M} of Proposition 2.2, we can assert that for every polynomial f and every x ∈ ℂ_{d} satisfying ∥x∥ ≤ 1 we have

|f(x)| ≤ sup_{z∈B_{d}} |f(z)| = ∥f∥_{∞} ≤ ∥f∥_{M}.

It follows that there is a unique complex homomorphism ω_{x} of A satisfying

ω_{x}(f) = f(x), f ∈ P.

For all g ∈ A we now have a natural continuous extension g̃ of g to the closed unit ball by setting

g̃(x) = ω_{x}(g), ∥x∥ ≤ 1.

x ↦ ω_{x} is a one-to-one continuous map of the closed ball in ℂ^{d} onto its range in σ(A). To see that it is surjective, let ω be an arbitrary element of σ(A). Then for every y ∈ ℂ^{d} we may consider the linear functional

ŷ(z) = ⟨z, y⟩ , z ∈ ℂ^{d}.

The map y ↦ ŷ is an antilinear mapping of ℂ^{d} onto the space of linear functions in P and we claim that ∥ŷ∥_{M} ≤ ∥y∥. Indeed, assuming that y ≠ 0, the linear function

u(x) = ŷ(x)/∥y∥ = ⟨x, y⟩ /∥y∥

is part of a system of coordinates for ℂ^{d}. Proposition 2.6 implies ∥u∥_{M} ≤ 1, hence ∥ŷ∥_{M} ≤ ∥y∥. Thus, y ↦ ω(ŷ) defines an antilinear functional on ℂ^{d} satisfying

|ω(ŷ)| ≤ ∥ŷ∥_{M} ≤ ∥y∥, y ∈ ℂ^{d}.

It follows that there is a unique vector x in the unit ball of ℂ^{d} such that

ω(ŷ) = ⟨x, y⟩ , y ∈ ℂ^{d}.

Thus, ω(f) = ω_{x}(f) on every linear functional f. Since both ω and ω_{x} are continuous unital homomorphisms of A, since P is the algebra generated by the linear functions and the constants, and since P is dense in A, it follows that ω = ω_{x}, and the claim is proved.

Thus we have identified the maximal ideal space of A with the closed unit ball in ℂ^{d}. From the elementary theory of commutative Banach algebras we deduce that for every f in A,

lim_{n→∞} ∥f^{n}∥_{M}^{1/n} = r(f) = sup{|ω(f)| : ω ∈ σ(A)} = sup{|f̃(x)| : ∥x∥ ≤ 1} = ∥f∥_{∞},

completing the proof of Proposition 2.12 □

The realization of the d-shift as a d-tuple of multiplication operators on the function space H^{2} is not always convenient for making computations. We require the following realization of (S_{1}, …, S_{d}) as “creation” operators on the symmetric Fock space F_{+}(E).

Proposition 2.13. Let e_{1}, … e_{d} be an orthonormal basis for a Hilbert space E of dimension d. Define operators A_{1}, …, A_{d} on F_{+}(E) by

A_{i}ξ = e_{i}ξ, ξ ∈ F_{+}(E)

where e_{i}ξ denotes the projection of e_{1}⊗ξ ∈ F(E) to the symmetric subspace F_{+}(E). Let z_{1}, …, z_{d} be the system of orthogonal coordinates z_{i}(x) = ⟨x, e_{i}⟩, 1 ≤ i ≤ d. Then there is a unique unitary operator W : H^{2} → F_{+}(E) such that W (1) = 1 and

(2.14)  W (z_{i_{1}} … z_{i_{n}}) = e_{i_{1}} … e_{i_{n}},  n ≥ 1, i_{k} ∈ {1, 2, …, d}.

In particular, the d-tuple of operators (A_{1}, …, A_{d}) is unitarily equivalent to the d-shift.

proof. For every x ∈ E satisfying ∥x∥ < 1 define an element v_{x} ∈ F_{+}(E) by

v_{x} = 1 ⊕ x ⊕ x^{2} ⊕ x^{3} ⊕ ….

it is obvious that ∥v_{x}∥^{2} = (1 − ∥x∥^{2})^{−1} and, more generally,

⟨v_{x}, v_{y}⟩ = (1 − ⟨x, y⟩)^{−1}, ∥x∥, ∥y∥ < 1.

F_{+}(E) is spanned by the set {v_{x} : ∥x∥ < 1}.

Let {u_{x} : ∥x∥ < 1} be the set of functions in H^{2} defined in (1.11), and let ∗ be the unique conjugation of E defined by e_{i}^{∗} = e_{i}, that is,

(a_{1}e_{1} + ⋯ + a_{d}e_{d})^{∗} = ā_{1}e_{1} + ⋯ + ā_{d}e_{d}.

Then we have

⟨u_{x}, u_{y}⟩ = (1 − ⟨y, x⟩)^{−1} = (1 − ⟨x^{∗}, y^{∗}⟩)^{−1} = ⟨v_{x^{∗}}, v_{y^{∗}}⟩

for all x, y in the open unit ball of E. By Proposition 1.12 there is a unique unitary operator W : H^{2} → F_{+}(E) such that W (u_{x}) = v_{x^{∗}}, ∥x∥ < 1.

We have W (1) = W (u_{0}) = v_{0} = 1. Choose x ∈ E satifying ∥x∥ ≤ 1 and let f_{x} denote the linear functional on E defined by f_{x}(z) = ⟨z, x⟩. We have ∥f_{x}∥_{H^{2}} ≤ 1 and in fact ∥f_{x}^{n}∥_{H^{2}} ≤ 1 for every n = 0, 1, 2, . . .. Hence for every 0 ≤ r < 1 and every z ∈ B_{d} we have

u_{rx}(z) = (1 − ⟨z, rx⟩)^{−1} = ∑_{n=0}^{∞} r^{n} ⟨z, x⟩^{n} = ∑_{n=0}^{∞} r^{n}f_{x}^{n}(z) ∈ H^{2}.

Similarly,

v_{rx^{∗}} = ∑_{n=0}^{∞} r^{n}(x^{∗})^{n} ∈ F_{+}(E).

Setting W (u_{rx}) equal to v_{rx^{∗}} and comparing coefficients of r^{n} we obtain

W (f_{x}^{n}) = (x^{∗})^{n}

for every n = 0, 1, . . .. It follows that

(2.15)  W (f_{x_{1}}f_{x_{2}} … f_{x_{n}}) = x_{1}^{∗}x_{2}^{∗} … x_{n}^{∗}

for every x_{1}, x_{2}, …, x_{n} ∈ E. Indeed, setting

L(x_{1}, x_{2}, …, x_{n}) = W (f_{x_{1}}f_{x_{2}} … f_{x_{n}})
R(x_{1}, x_{2}, …, x_{n}) = x_{1}^{∗}x_{2}^{∗} … x_{n}^{∗}

for x_{1}, x_{2}, …, x_{n} ∈ E, we see that both L and R are symmetric n-antilinear mappings which agree when x_{1} = x_{2} = ⋯ = x_{n} ∈ B_{d}. Hence L = R and (2.15) follows. We obtain (2.14) by taking x_{k} = e_{i_{k}} in (2.15).

(2.14) obviously implies W S_{i} = A_{i}W for i = 1, …, d, so that the d-tuples (S_{1}, …, S_{d}) and (A_{1}, …, A_{d}) are unitarily equivalent □

3. von Neumann’s inequality and the sup norm.

Definition 3.1. A d-contraction is a d-tuple of operators T̄ = (T_{1}, …, T_{d}) acting on a common Hilbert space H which commute with each other and satisfy

∥T_{1}ξ_{1} + ⋯ + T_{d}ξ_{d}∥^{2} ≤ ∥ξ_{1}∥^{2} + ⋯ + ∥ξ_{d}∥^{2},

for every ξ_{1}, …, ξ_{d} ∈ H.

Remark 3.2. We make frequent use of the following observation. For operators T_{1}, …, T_{d} on a common Hilbert space H, the following are equivalent.

(1) ∥T_{1}ξ_{1} + ⋯ + T_{d}ξ_{d}∥^{2} ≤ ∥ξ_{1}∥^{2} + ⋯ + ∥ξ_{d}∥^{2} for all ξ_{1}, …, ξ_{d} ∈ H.

(2) T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗} ≤ 1. To see this let d · H denote the direct sum of d copies of H, and let T̄ ∈ B(d · H, H) be the operator defined by T̄(ξ_{1}, …, ξ_{d}) = T_{1}ξ_{1} + ⋯ + T_{d}ξ_{d}. A simple computation shows that the adjoint T̄^{∗} : H → d · H is given by

T̄^{∗}ξ = (T_{1}^{∗}ξ, …, T_{d}^{∗}ξ).

Thus T̄T̄^{∗} is the operator in B(H) given by T̄T̄^{∗} = T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗}. The equivalence of (1) and (2) follows.

Notice that the d-shift S̄ = (S_{1}, …, S_{d}) acting on H_{d}^{2} is a d-contraction. Perhaps the most natural generalization of von Neumann’s inequality for d-dimensional operator theory would make the following assertion. Let T̄ = (T_{1}, …, T_{d}) be a d-contraction and let f = f(z_{1}, …, z_{d}) be a polynomial in d complex variables z_{1}, …, z_{d}. Then

∥f(T_{1}, …, T_{d})∥ ≤ sup_{∥z∥≤1} |f(z_{1}, …, z_{d})|.

In this section we show that this inequality fails rather spectacularly for the d-shift, in that there is no constant K for which

∥f(S_{1}, …, S_{d})∥ ≤ K sup_{∥z∥≤1} |f(z_{1}, …, z_{d})|,

holds for all polynomials f. It follows that the multiplier algebra M is a proper subalgebra of H^{∞}. Indeed, we exhibit continuous functions

f : {z ∈ ℂ^{d} : ∥z∥ ≤ 1} → ℂ

which are analytic in the interior of the unit ball and which do not belong to M.

We will establish the appropriate version of von Neumann’s inequality for dimension d ≥ 2 in section 8.

Theorem 3.3. Assume d ≥ 2. Let c_{0}, c_{1}, … be a sequence of complex numbers having the properties

∑_{n=0}^{∞} |c_{n}| = 1, (i)

∑_{n=0}^{∞} |c_{n}|^{2}n^{(d−1)/2} = ∞, (ii)

and define a function f(z_{1}, …, z_{d}) for |z_{1}|^{2} + ⋯ + |z_{d}|^{2} ≤ 1 as follows

(3.3)  f(z_{1}, …, z_{d}) = ∑_{n=0}^{∞} \frac{c_{n}}{s^{n}} (z_{1}z_{2} … z_{d})^{n},

where s denotes the sup norm

(3.4)  s = sup_{|z_{1}|^{2}+···+|z_{d}|^{2}≤1} |z_{1}z_{2} … z_{d}| = \sqrt{\frac{1}{d^{d}}} .

Then the power series (3.3) converges uniformly over the closed unit ball to a function f satisfying ∥f∥_{∞} ≤ 1. The restriction of f to B_{d} does not belong to H^{2}. Letting f_{0}, f_{1}, f_{2}, … be the sequence of Taylor polynomials

f_{N}(z_{1}, …, z_{d}) = ∑_{n=0}^{N} \frac{c_{n}}{s^{n}} (z_{1}z_{2} … z_{d})^{n},

then we have ∥f_{N}∥_{∞} ≤ 1 for every N while

(3.5)  lim_{N→∞} ∥f_{N}(S_{1}, …, S_{d})∥ = lim_{N→∞} ∥f_{N}∥_{M} = ∞.

Remarks. It is clear that the function f belongs to the “ball algebra”, that is the closure in the sup norm ∥ · ∥_{∞} of the algebra of polynomials. On the other hand, f does not belong to the multiplier algebra M, and in particular the inclusion M ⊊ H^{∞} is proper.

Note too that it is a simple matter to give explicit examples of sequences c_{0}, c_{1}, … satisfying conditions (i) and (ii). For example, let S be any infinite subset of the nonnegative integers which is sparse enough so that

∑_{n∈S} \frac{1}{n^{(d−1)/4}} < ∞.

If we set c_{n} = 1/n^{(d−1)/4} if n ∈ S and c_{n} = 0 otherwise, then we obviously have (ii) because S is infinite, and (i) can also be achieved after multiplying the sequence by a suitable positive constant.

proof. The formula (3.4) for the sup norm,

s = d^{−d/2}

follows from the elementary fact that

(|z_{1}|^{2}|z_{2}|^{2} … |z_{d}|^{2})^{1/d} ≤ (|z_{1}|^{2} + |z_{2}|^{2} + ⋯ + |z_{d}|^{2})/d

with equality iff |z_{1}| = |z_{2}| = ⋯ = |z_{d}|.

Let p be the homogeneous polynomial p(z_{1}, …, z_{d}) = z_{1}z_{2} … z_{d}. Then for every n = 0, 1, 2, … we have

∥p^{n}∥_{∞} = ∥p∥_{∞}^{n} = d^{−nd/2} = s^{n}.

It follows that the power series (3.3)

∑_{n=0}^{∞} \frac{c_{n}}{s^{n}} p(z_{1}, …, z_{d})^{n}

converges uniformly over the unit ball to f. Thus it remains to establish the condition (3.5).

Now for any polynomial g in the d complex variables z_{1}, …, z_{d} we have

∥g(S_{1}, …, S_{d})∥ ≥ ∥g(S_{1}, …, S_{d})1∥_{H^{2}} = ∥g∥_{H^{2}}.

Thus it suffices to show that the sequence of Taylor polynomials f_{0}, f_{1}, f_{2}, … defined by the partial sums of the series diverges in the H^{2} norm, that is

(3.6)  sup_{N} ∥f_{N}∥_{H^{2}}^{2} = ∑_{n=0}^{∞} |c_{n}|^{2} \frac{∥p^{n}∥_{H^{2}}^{2}}{∥p^{n}∥_{∞}^{2}} = ∞.

In order to establish (3.6) we will show that there is a positive constant A such that

(3.7)  ∥p^{n}∥_{H^{2}}^{2} ≥ Ad^{−nd}n^{(d−1)/2},

for all n = 1, 2, . . .. In view of the fact that ∥p^{n}∥_{∞}^{2} = d^{−nd} and the series ∑_{n} |c_{n}|^{2}n^{(d−1)/2} diverges, (3.6) will follow.

The estimate (3.7) is based on the following computation. Since the result is a statement about certain norms in the symmetric Fock space over ℂ^{d}, it is likely that the result of Lemma 3.8 can be found in the literature. Since we are not aware of an appropriate reference and since the estimate (3.7) depends essentially on these formulas, we have provided the details.

Lemma 3.8. Let e_{1}, e_{2}, …, e_{d} be an orthonormal basis for E = ℂ^{d}. Then for every d-tuple of nonnegative integers k = (k_{1}, …, k_{d}) we have

∥e_{1}^{k_{1}}e_{2}^{k_{2}} … e_{d}^{k_{d}}∥_{E^{|k|}}^{2} = \frac{k_{1}!k_{2}! … k_{d}!}{|k|!}

where |k| = k_{1} + k_{2} + ⋯ + k_{d}.

Remark. Regarding notation, we have written e_{1}^{k_{1}}e_{2}^{k_{2}} … e_{d}^{k_{d}} for the projection of the vector

e_{1}^{⊗k_{1}} ⊗ e_{2}^{⊗k_{2}} ⊗ ⋯ ⊗ e_{d}^{⊗k_{d}} ∈ E^{⊗|k|}

to the symmetric subspace E^{|k|} ⊆ E^{⊗|k|}.

proof. For y_{1}, …, y_{p} ∈ E = ℂ^{d} we write y_{1}y_{2} … y_{p} for the projection of y_{1} ⊗ y_{2} ⊗ ⋯ ⊗ y_{p} ∈ E^{⊗p} to the symmetric subspace E^{p}. Fixing a ∈ E, and p ≥ 1 we have an associated “creation operator” A : E^{p−1} → E^{p} defined by

A(x_{1}x_{2} … x_{p−1}) = ax_{1}x_{2} … x_{p−1}, x ∈ E.

We claim first that for p ≥ 1, the adjoint A^{∗} : E^{p} → E^{p−1} is given by

(3.9)  A^{∗}(y_{1}y_{2} … y_{p}) = \frac{1}{p} ∑_{k=1}^{p} ⟨y_{k}, a⟩ y_{1} … ŷ_{k} … y_{p}

where ŷ_{k} means that the term y_{k} is missing from the symmetric tensor product. Indeed, if ζ denotes the right side of (3.9) then for every x ∈ E we have

⟨ζ, x^{p−1}⟩ = \frac{1}{p} ∑_{k=1}^{p} ⟨y_{1}, x⟩ … ⟨y_{k−1}, x⟩ ⟨y_{k}, a⟩ ⟨y_{k+1}, x⟩ … ⟨y_{p}, x⟩
= ⟨y_{1} ⊗ ⋯ ⊗ y_{p}, p^{−1}(a ⊗ x^{p−1} + x ⊗ a ⊗ x^{p−2} ⋯ + x^{p−1} ⊗ a)⟩ .

Since

p^{−1}(a ⊗ x^{p−1} + x ⊗ a ⊗ x^{p−2} ⋯ + x^{p−1} ⊗ a) = ax^{p−1} ∈ E^{p},

the right side of the preceding formula becomes

⟨y_{1} ⊗ ⋯ ⊗ y_{p}, ax^{p−1}⟩ = ⟨y_{1}y_{2} … y_{p}, ax^{p−1}⟩ = ⟨A^{∗}(y_{1}y_{2} … y_{p}), x^{p−1}⟩ .

(3.9) now follows because E^{p−1} is spanned by vectors of the form x^{p−1}, x ∈ E.

To prove Lemma 3.8 we proceed by induction on the total degree |k|. The formula is obvious for |k| = 0. Assuming that |k| ≥ 1 and that the formula has been established for total degree |k| − 1 then we may assume (after relabelling the basis vectors e_{1}, …, e_{d} if necessary) that k_{1} ≥ 1.

Taking a = e_{1} in (3.9) and noting that ⟨e_{1}, e_{1}⟩ = 1 and ⟨e_{1}, e_{j}⟩ = 0 if j = 2, …, d, we find that

A^{∗}(e_{1}^{k_{1}}e_{2}^{k_{2}} … e_{d}^{k_{d}}) = \frac{k_{1}}{|k|}e_{1}^{k_{1}−1}e_{2}^{k_{2}} … e_{d}^{k_{d}},

hence

⟨e_{1}^{k_{1}} … e_{d}^{k_{d}}, e_{1}^{k_{1}} … e_{d}^{k_{d}}⟩ = ⟨A^{∗}(e_{1}^{k_{1}} … e_{d}^{k_{d}}), e_{1}^{k_{1}−1}e_{2}^{k_{2}} … e_{d}^{k_{d}}⟩

= \frac{k_{1}}{|k|}∥e_{1}^{k_{1}−1}e_{2}^{k_{2}} … e_{d}^{k_{d}}∥^{2}.

The required formula now follows from the induction hypothesis □

Setting k_{1} = k_{2} = ⋯ = k_{d} = n in Lemma 3.8, we obtain

∥(e_{1}e_{2} … e_{d})^{n}∥_{E^{nd}}^{2} = \frac{(n!)^{d}}{(nd)!}.

The right side is easily estimated using Stirling’s formula

n! ∼ \sqrt{2π} n^{n+1/2} e^{−n}

and after obvious cancellations we find that

\frac{(n!)^{d}}{(nd)!} ∼ (\frac{2π^{d−1}}{d})^{1/2}d^{−nd}n^{(d−1)/2}.

In order to deduce (3.7) from the latter, choose an orthonormal basis e_{1}, e_{2}, …, e_{d} for ℂ^{d} so that

z_{k}(x) = ⟨x, e_{k}⟩ , k = 1, 2, …, d.

Then

∥(z_{1}z_{2} … z_{d})^{n}∥_{H^{2}}^{2} = ∥(e_{1}e_{2} … e_{d})^{n}∥_{E^{nd}}^{2}

and (3.7) follows after choosing A to be a positive number appropriately smaller than \sqrt{2π^{d−1}/d}. That completes the proof of Theorem 3.3 □

Remark 3.10. We recall that a d-tuple of commuting operators T̄ = (T_{1}, T_{2}, …, T_{d}) on a Hilbert space H is said to be subnormal if there is a commuting d-tuple of normal operators N̄ = (N_{1}, N_{2}, …, N_{d}) on a larger Hilbert space K ⊇ H such that

T_{k} = N_{k} ↾_{H}, k = 1, 2, …, d.

The one-dimensional unilateral shift can be extended to a unitary operator on a larger space. That situation is unique to dimension 1, as we have

Corollary 1. For every d ≥ 2 the d-shift is not subnormal.

proof. In Proposition 2.12 we identified the maximal ideal space of the unital Banach algebra generated by the d-shift with the closed unit ball in ℂ^{d}. In particular, for every polynomial f the spectral radius of f(S_{1}, …, S_{d}) is given by

r(f(S_{1}, …, S_{d})) = sup_{|z_{1}|^{2}+···+|z_{d}|^{2}≤1} |f(z_{1}, …, z_{d})|.

If the d-shift were subnormal then f(S_{1}, …, S_{d}) would be a subnormal operator for every polynomial f, hence its norm would equal its spectral radius [22, Problem 162], contradicting Theorem 3.3. □

The two most common Hilbert spaces associated with the unit ball B_{d} arise from measures. These are the spaces H^{2}(∂B_{d}) associated with normalized surface measure on the boundary of B_{d} and the space H^{2}(B_{d}) associated with normalized volume measure on B_{d} [37]. It is reasonable to ask if the space H^{2} can be associated with some measure on ℂ^{d}. The answer is no because that would imply that the d-shift is subnormal, contradicting Corollary 1. The details are as follows.

Corollary 2. There is no positive measure μ on ℂ^{d}, d ≥ 2, with the property that

∥f∥_{H^{2}}^{2} = ∫_{ℂ^{d}} |f(z)|^{2} dμ(z)

for every polynomial f.

proof. Suppose that such a measure μ did exist. μ must be a probability measure because ∥1∥_{H^{2}} = 1, and it must have finite moments of all orders.

We claim that μ must have compact support. Indeed, if f is any linear functional on ℂ^{d} of the form f(x) = ⟨x, e⟩ where e is a unit vector of ℂ^{d} then by Lemma 3.8 we have

∥f^{n}∥_{H^{2}}^{2} = ∥e^{n}∥_{E^{n}}^{2} = 1

for every n = 1, 2, . . .. Hence

∫_{ℂ^{d}} |f(z)|^{2n} dμ(z) = 1.

Taking 2nth roots we find that the function f has norm 1 when it is considered an element in the space L^{p}(ℂ^{d}, μ) for p = 2, 4, 6, . . .. Letting X be the closed support of the measure μ we find that

sup_{z∈X} |f(z)| = lim_{n→∞}(∫_{ℂ^{d}} |f|^{2n} dμ)^{1/2n} = 1.

This proves that for every z ∈ X and e in the unit ball of ℂ^{d} we have

| ⟨z, e⟩ | ≤ 1,

thus X must be contained in the closed unit ball of ℂ^{d}.

Now we simply view the d-shift as a d-tuple of multiplication operators in the space L^{2}(μ). Here, S_{k} is multiplication by z_{k} acting on the closure (in L^{2}(μ)) of the space of polynomials. This d-tuple (S_{1}, …, S_{d}) is obviously subnormal, contradicting Corollary 1 above □

Remark 3.11. In the conventional approach to dilation theory one seeks normal dilations for operators or sets of operators. Theorem 3.3 implies that this approach is inappropriate for d-contractions and the unit ball of ℂ^{d} in dimension greater than one. Indeed, if (N_{1}, …, N_{d}) is a d-tuple of mutually commuting normal operators whose joint spectrum is contained in the closed unit ball of ℂ^{d} then for every polynomial f ∈ P we have

∥f(N_{1}, …, N_{d})∥ ≤ sup_{z∈B_{d}} |f(z)|.

Since Theorem 3.3 implies that there are polynomials f for which the inequality

∥f(S_{1}, …, S_{d})∥ ≤ sup_{z∈B_{d}} |f(z)|

fails, one cannot obtain such operators f(S_{1}, …, S_{d}) by compressing f(N_{1}, …, N_{d}) to any subspace. Thus the d-shift cannot be dilated to a normal d-tuple having its spectrum in the closed unit ball.

4. Maximality of the H^{2} norm.

The purpose of this section is to show that in every dimension d = 1, 2, … the H^{2} norm is distinguished among all Hilbert seminorms defined on the space P of polynomials by being the largest Hilbert seminorm which is appropriate for operator theory on the unit ball of ℂ^{d}. As a consequence, we show that the function space H^{2} is contained in every other Hilbert space of analytic functions on the open unit ball which has these natural properties.

Definition 4.1. Let z_{1}, …, z_{d} be a system of coordinate functions on ℂ^{d}. A Hilbert seminorm ∥ · ∥ defined on the space P of all polynomials is said to be contractive if for every a ∈ ℂ and every f_{1}, …, f_{d} ∈ P we have

∥a1 + z_{1}f_{1} + … z_{d}f_{d}∥^{2} ≤ |a|^{2} + ∥f_{1}∥^{2} + ⋯ + ∥f_{d}∥^{2}.

Remarks. Proposition 2.6 asserts that the H^{2} norm is a contractive norm on P. From Proposition 4.2 below it follows that the Hilbert norms defined on P by both H^{2}(B_{d}) and H^{2}(∂B_{d}) are contractive norms.

It is a simple exercise to show that if a Hilbert seminorm ∥ · ∥ is contractive relative to one system of coordinates z_{1}, …, z_{d} then it is contractive relative to every system of coordinates. Thus the definition of contractive seminorm depends only on the structure of ℂ^{d} as a d-dimensional Hilbert space.

Notice too that if ∥ · ∥ is any contractive seminorm then for any system of coordinate functions z_{1}, …, z_{d} the multiplication operators (M_{z_{1}}, …, M_{z_{d}}) give rise to a d-contraction acting on the Hilbert space obtained by completing P in this seminorm. Indeed, we have the following somewhat more concrete characterization of contractive Hilbert seminorms.

Proposition 4.2. Let ∥ · ∥ be an arbitrary Hilbert seminorm on P, let H be the inner product space defined by ∥ · ∥, and let P_{0} be the maximal ideal in P consisting of all polynomials f such that f(0) = 0. Then ∥ · ∥ is a contractive seminorm iff the following two conditions are satisfied

(1) 1 ⊥ P_{0} in the space H, and

(2) For some system of coordinate functions z_{1}, …, z_{d} the multiplication operators (M_{z_{1}}, …, M_{z_{d}}) define a d-contraction on H.

proof. Once one notes that the most general element of P_{0} is a sum of the form z_{1}f_{1} + ⋯ + z_{d}f_{d} with f_{1}, …, f_{d} ∈ P, the argument is straightforward □

We collect the following observation, which asserts that condition (2) alone is enough in the presence of minimal symmetry.

Corollary. For every λ in the circle group {z ∈ ℂ : |z| = 1} and every f ∈ P set f_{λ}(z) = f(λz), z ∈ ℂ^{d}. Let ∥ · ∥ be a Hilbert seminorm on P which satisfies ∥f_{λ}∥ = ∥f∥ for every f ∈ P and every λ, such that for some system of coordinate functions z_{1}, …, z_{d}, the multiplication operators M_{z_{1}}, …, M_{z_{d}} give rise to a dcontraction acting on the Hilbert space H obtained from ∥ · ∥.

Then ∥ · ∥ is a contractive seminorm.

proof. We show that the symmetry hypothesis implies condition (1) of Proposition 4.2. For every λ in the unit circle we can define a unitary operator U_{λ} uniquely on H by setting

U_{λ}f = f_{λ}, f ∈ P.

It is obvious that U is a unitary representation of the circle group on H. Moreover, if f is a homogeneous polynomial of degree n = 0, 1, … then we have

U_{λ}f = λ^{n}f

for all λ. Thus for the inner product ⟨·, ·⟩ associated with ∥ · ∥ we have

⟨f, 1⟩ = ⟨U_{λ}f, U_{λ}1⟩ = λ^{n} ⟨f, 1⟩ ,

so that if n ≥ 1 then ⟨f, 1⟩ = 0. It follows that 1 ⊥ P_{0}, as required □

Following is the main result of this section.

Theorem 4.3. Let ∥·∥ be any contractive Hilbert seminorm on P. Then for every f ∈ P we have

∥f∥ ≤ k∥f∥_{H^{2}}

where k = ∥1∥. In particular, the H^{2} norm is the largest contractive Hilbert seminorm which assigns norm 1 to the constant polynomial f = 1.

In particular, we see that the Hilbert norms arising from the “Hardy” space H^{2}(∂B_{d}) and the “Bergman” space H^{2}(B_{d}) are both dominated by ∥·∥_{H^{2}}. Indeed, we have the following inclusions of the corresponding Hilbert spaces of analytic functions in the open ball B_{d}

H^{2} ⊆ H^{2}(∂B_{d}) ⊆ H^{2}(B_{d}),

where both inclusion maps are compact operators of norm 1. Since we do not require the latter assertion, we omit the proof. However, note that of these three function spaces, H^{2} is the only one that does not contain H^{∞}, and it is the only one of the three for which the d-contraction defined by the multiplication operators (M_{z_{1}}, …, M_{z_{d}}) fails to be subnormal.

Remark 4.4. Every d-contraction (T_{1}, …, T_{d}) in B(H) gives rise to a normal completely positive map P on B(H) by way of

P (A) = T_{1}AT_{1}^{∗} + ⋯ + T_{d}AT_{d}^{∗}, A ∈ B(H).

Because of Remark 3.2 we have P (1) = T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗} ≤ 1, and in fact the sequence A_{n} = P^{n}(1) is decreasing: A_{0} = 1 ≥ A_{1} ≥ A_{2} ≥ ⋯ ≥ 0. Thus

A_{∞} = lim_{n→∞} P^{n}(1)

exists as a limit in the strong operator topology and satisfies 0 ≤ A_{∞} ≤ 1. A d-contraction T̄ = (T_{1}, …, T_{d}) is called null if A_{∞} = 0. Notice that if the row norm of T̄ is less than 1, i.e., T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗} ≤ r1 for some 0 < r < 1, then ∥P ∥ = ∥P (1)∥ ≤ r < 1 and hence T̄ is a null d-contraction.

For the proof of Theorem 4.3 we require an operator theoretic result which relates closely to the material of section 6. Gelu Popescu has pointed out that the operator L is related to his Poisson kernel operator K_{r} of [35, section 8], when r = 1.

Theorem 4.5. Let (T_{1}, …, T_{d}) be a d-contraction on a Hilbert space H, define the operator

Δ = (1 − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗})^{1/2},

and the subspace K = \overline{ΔH}. Let E be a d-dimensional Hilbert space and let

F_{+}(E) = ℂ ⊕ E ⊕ E^{2} ⊕ …

be the symmetric Fock space over E.

Then for every orthonormal basis e_{1}, …, e_{d} for E there is a unique bounded operator L : F_{+}(E) ⊗ K → H satisfying L(1 ⊗ ξ) = Δξ and

L(e_{i_{1}}e_{i_{2}} … e_{i_{n}} ⊗ ξ) = T_{i_{1}}T_{i_{2}} … T_{i_{n}}Δξ

for every i_{1}, …, i_{n} ∈ {1, 2, …, d}, n = 1, 2, . . .. In general we have ∥L∥ ≤ 1, and if (T_{1}, …, T_{d}) is a null d-tuple, then L is a coisometry: LL^{∗} = 1_{H}.

proof. If there is a bounded operator L satisfying the stated condition then it is obviously unique because F_{+}(E) is spanned by the set of vectors

{1, e_{i_{1}}, e_{i_{2}}e_{i_{3}}, e_{i_{4}}e_{i_{5}}e_{i_{6}}, ⋯: i_{k} ∈ {1, 2, …, d}, k = 1, 2, … }.

We define L by exhibiting its adjoint. That is, we will exhibit an operator A : H → F(E) ⊗ K, F(E) denoting the full Fock space over E, and we will show that ∥A∥ ≤ 1 and

(4.6)  ⟨L(ζ), η⟩ = ⟨ζ, A(η)⟩

for ζ of the form 1 ⊗ ξ or e_{i_{1}}e_{i_{2}} … e_{i_{n}} ⊗ ξ for ξ ∈ K. At that point we can define L to be the adjoint of P_{+}A, P_{+} denoting the projection of F(E) onto its subspace F_{+}(E).

For every η ∈ H, we define Aη as a sequence of vectors (ζ_{0}, ζ_{1}, ζ_{2}, …) where ζ_{n} ∈ E^{⊗n} ⊗ K is defined by

ζ_{n} = ∑_{i_{1},...,i_{n}=1}^{d} e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}} ⊗ ΔT_{i_{n}}^{∗} … T_{i_{1}}^{∗}η

for n ≥ 1 and ζ_{0} = 1 ⊗ Δη. Notice that since T_{1}^{∗}, …, T_{d}^{∗} commute, ζ_{n} actually belongs to the symmetric subspace E^{n} ⊗ K. We claim first that

∑_{n=0}^{∞} ∥ζ_{n}∥^{2} ≤ ∥η∥^{2}

so that in fact A maps into F(E) ⊗ K and is a contraction. Indeed, we have

∥ζ_{n}∥^{2} = ∑_{i_{1},...,i_{n}=1}^{d} ∥ΔT_{i_{n}}^{∗} … T_{i_{1}}^{∗}η∥^{2} = ∑_{i_{1},...,i_{n}=1}^{d} ⟨T_{i_{1}} … T_{i_{n}}Δ^{2}T_{i_{n}}^{∗} … T_{i_{1}}^{∗}η, η⟩ .

Let P (A) = T_{1}AT_{1}^{∗} + ⋯ + T_{d}AT_{d}^{∗} be the completely positive map of Remark 4.4. Noting that Δ^{2} = 1 − P (1) we find that

∑_{i_{1},...,i_{n}=1}^{d} T_{i_{1}} … T_{i_{n}}Δ^{2}T_{i_{n}}^{∗} … T_{i_{1}}^{∗} = P^{n}(1 − P (1)) = P^{n}(1) − P^{n+1}(1),

and hence

∥ζ_{n}∥^{2} = ⟨P^{n}(1)η, η⟩ − ⟨P^{n+1}(1)η, η⟩ .

The series ∥ζ_{0}∥^{2} + ∥ζ_{1}∥^{2} + … therefore telescopes and we are left with

(4.7)  ∑_{n=0}^{∞} ∥ζ_{n}∥^{2} = ∥η∥^{2} − ⟨A_{∞}η, η⟩ ≤ ∥η∥^{2},

where A_{∞} is the positive contraction A_{∞} = lim_{n→∞} P^{n}(1) of Remark 4.4.

We now verify (4.6) for ζ of the form ζ = e_{j_{1}} … e_{j_{n}} ⊗ ξ for n ≥ 1, j_{1}, …, j_{n} ∈ {1, 2, …, d} and ξ ∈ K. We have

⟨e_{j_{1}} … e_{j_{n}} ⊗ ξ, Aη⟩ = ∑_{i_{1},...,i_{n}=1}^{d} ⟨e_{j_{1}} … e_{j_{n}} ⊗ ξ, e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}} ⊗ ΔT_{i_{n}}^{∗} … T_{i_{1}}^{∗}η⟩

= ∑_{i_{1},...,i_{n}=1}^{d} ⟨e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{n}}, e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}}⟩ ⟨ξ, ΔT_{i_{n}}^{∗} … T_{i_{1}}^{∗}η⟩

= ⟨ξ, ΔT_{j_{1}}^{∗} … T_{j_{1}}^{∗}η⟩ = ⟨T_{j_{1}} … T_{j_{n}}Δξ, η⟩ = ⟨L(ζ), η⟩ .

For ζ = 1 ⊗ ξ with ξ ∈ K we have

⟨1 ⊗ ξ, Aη⟩ = ⟨1 ⊗ ξ, 1 ⊗ Δη⟩ = ⟨ξ, Δη⟩ = ⟨Δξ, η⟩ ,

as required. If (T_{1}, …, T_{d}) is a null d-tuple, then it is clear from (4.7) that A is an isometry and hence L is a coisometry. □

proof of Theorem 4.3. Let H be the Hilbert space obtained by completing P in the seminorm ∥·∥. Choose an orthonormal basis e_{1}, …, e_{d} for E = ℂ^{d} and let z_{1}, …, z_{d} be the corresponding system of coordinate functions z_{i}(x) = ⟨x, e_{i}⟩, i = 1, …, d.

Since ∥ · ∥ is a contractive Hilbert seminorm the multiplication operators

T_{k} = M_{z_{k}}, k = 1, …, d

define a d-contraction (T_{1}, …, T_{d}) in B(H). Set

Δ = (1 − ∑_{k=1}^{d} T_{k}T_{k}^{∗})^{1/2},

let K = \overline{ΔH} be the closed range of Δ and let L : F_{+}(E)⊗K → H be the contraction defined in Theorem 4.5 by the conditions L(1 ⊗ ξ) = Δξ and, for n = 1, 2, …

(4.8)  L(e_{i_{1}} … e_{i_{n}} ⊗ ξ) = T_{i_{1}} … T_{i_{n}}Δξ,

ξ ∈ K, i_{1}, …, i_{n} ∈ {1, 2, …, d}.

The constant polynomial 1 ∈ P is represented by a vector v in H. We claim that Δv = v. Indeed, since ∥ · ∥ is a contractive seminorm, condition (1) of Proposition 4.2 implies that

v ⊥ T_{1}H + T_{2}H + ⋯ + T_{d}H,

and hence T_{k}^{∗}v = 0 for k = 1, …, d. It follows that

∥Δv∥^{2} = ⟨Δ^{2}v, v⟩ = ∥v∥^{2} − ∑_{k=1}^{d} ∥T_{k}^{∗}v∥^{2} = ∥v∥^{2},

and hence Δv = v because 0 ≤ Δ ≤ 1.

In particular, v = Δv ∈ \overline{ΔH} = K. Taking ξ = v in (4.8) we obtain

L(e_{i_{1}} … e_{i_{n}} ⊗ v) = T_{i_{1}} … T_{i_{n}}v.

Since v is the representative of 1 in H, T_{i_{1}} … T_{i_{n}}v is the representative of the polynomial z_{i_{1}} … z_{i_{n}} in H, and we have

L(e_{i_{1}} … e_{i_{n}} ⊗ v) = z_{i_{1}} … z_{i_{n}} ∈ H.

By Propostion 2.13 there is a unitary operator W : H^{2} → F_{+}(E) which carries 1 to 1 and carries z_{i_{1}} … z_{i_{n}} ∈ H^{2} to e_{i_{1}} … e_{i_{n}} ∈ F_{+}(E). Hence

L(W (z_{i_{1}} … z_{i_{n}}) ⊗ v) = z_{i_{1}} … z_{i_{n}}.

By taking linear combinations we find that for every polynomial f ∈ P

L(W f ⊗ v) = f

where, f on the left is considered an element of H^{2} and f on the right is considered an element of H. Since ∥L∥ ≤ 1 and W is unitary, we immediately deduce that

∥f∥_{H} ≤ ∥W f ⊗ v∥ = ∥f∥_{H^{2}} · ∥v∥_{H}.

Theorem 4.3 follows after noting that ∥v∥_{H} = ∥1∥_{H} □

Remarks. In particular, the H^{2} norm is the largest Hilbert seminorm ∥ · ∥ on the space P of all polynomials which is contractive and is normalized so that ∥1∥ = 1.

The following result gives a precise sense in which the space H^{2} will be present in any function space that is appropriate for doing operator theory relative to the unit ball in ℂ^{d}.

Corollary. Let F be a linear space of analytic functions defined on the open unit ball in ℂ^{d}, d = 1, 2 …, which contains all polynomials and is a Hilbert space relative to some Hilbert norm with the following two properties:

(1) The constant functions are orthogonal to the space P_{0} of polynomials which vanish at 0

(2) For some orthonormal set of linear functionals f_{1}, …, f_{d} on ℂ^{d} the multiplication operators (M_{f_{1}}, …, M_{f_{d}}) define a d-contraction on this Hilbert space.

Then F contains all functions in H^{2} and the inclusion map of H^{2} into F is a bounded operator.

Remark. By an orthonormal set of linear functionals we of course mean a set of linear functionals of the form f_{k}(x) = ⟨x, e_{k}⟩ where e_{1}, …, e_{d} is an orthonormal basis for ℂ^{d}.

proof. Let H be the subspace of F obtained by closing the space of polynomials in the norm ∥ · ∥ of F. Since f ∈ P ↦ ∥f∥ is a contractive Hilbert norm on polynomials, Theorem 4.3 implies that ∥f∥ ≤ k∥f∥_{H^{2}} for every f ∈ P, where k = ∥1∥ > 0. Thus H^{2} is contained in H ⊆ F and the inclusion of H^{2} in F is of norm at most k □

We will make use of the following extremal property of the H^{2} norm below.

Corollary 4.9. Let ∥·∥ be a contractive Hilbert seminorm on P satisfying ∥1∥ = 1 and let z_{1}, …, z_{d} be a system of orthogonal coordinate functions for E = ℂ^{d}. Then for every n = 1, 2, … we have

(4.10)  ∑_{i_{1},...,i_{n}=1}^{d} ∥z_{i_{1}}z_{i_{2}} … z_{i_{n}}∥^{2} ≤ \frac{(n + d − 1)!}{n!(d − 1)!},

with equality holding iff ∥f∥ = ∥f∥_{H^{2}} for every polynomial f of degree at most n.

proof. Let {e_{1}, …, e_{d}} be an orthonormal basis for a d-dimensional Hilbert space E. We consider the projection P_{n} ∈ B(E^{⊗n}) of the full tensor product onto its symmetric subspace E^{n}. Since ∥ · ∥ is a contractive seminorm, Theorem 4.3 implies that for all i_{1}, …, i_{n} we have

∥z_{i_{1}} … z_{i_{n}}∥ ≤ ∥z_{i_{1}} … z_{i_{n}}∥_{H^{2}} = ∥P_{n}(e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}})∥

and hence

∑_{i_{1},...,i_{n}=1}^{d} ∥z_{i_{1}} … z_{i_{n}}∥^{2} ≤ ∑_{i_{1},...,i_{n}=1}^{d} ∥P_{n}(e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}})∥^{2}.

Since {e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}} : 1 ≤ i_{1}, …, i_{n} ≤ d} is an orthonormal basis for E^{⊗n} the term on the right is trace (P_{n}) = dim(E^{n}), and (4.10) follows from the computation of the dimension of E^{n} in A.5.

Let P_{n} denote the subspace of H^{2} consisting of homogeneous polynomials of degree n, and let Q_{n} be the projection of H^{2} on P_{n}. The preceding observations imply that if A is any operator on H^{2} which is supported in P_{n} in the sense that A = Q_{n}AQ_{n} then the trace of A is given by

(4.11)  trace (A) = ∑_{i_{1},...,i_{n}=1}^{d} ⟨Az_{i_{1}} … z_{i_{n}}, z_{i_{1}} … z_{i_{n}}⟩_{H^{2}} .

Now fix n and suppose equality holds in (4.10). Since ∥ · ∥ is a contractive Hilbert seminorm satisfying ∥1∥ = 1, Theorem 4.3 implies that ∥f∥ ≤ ∥f∥_{H^{2}} for every f ∈ P, and hence there is a unique operator H ∈ B(H^{2}) satisfying

⟨f, g⟩ = ⟨Hf, g⟩_{H^{2}} , f, g ∈ P

and one has 0 ≤ H ≤ 1. Considering the compression Q_{n}HQ_{n} of H to P_{n} we see from (4.11) that

trace (Q_{n}HQ_{n}) = dim(E^{n}) = trace (Q_{n}).

Since Q_{n} − Q_{n}HQ_{n} ≥ 0 and the trace is faithful, we conclude that Q_{n}HQ_{n} = Q_{n}, and since H is a positive contraction it follows that Hf = f for every f ∈ P_{n}.

We claim that Hf = f for every f ∈ P_{k} and every k = 0, 1, …, n. To see that, choose a linear functional z ∈ P satisfying ∥z∥_{H^{2}} = 1. Since ∥ · ∥ is a contractive seminorm we have ∥z · f∥ ≤ ∥f∥ for every f ∈ P and in particular we have ∥z^{n}∥ = ∥z^{n−k}z^{k}∥ ≤ ∥z^{k}∥. Thus

⟨Hz^{k}, z^{k}⟩_{H^{2}} = ∥z^{k}∥^{2} ≥ ∥z^{n}∥^{2} = ∥z^{n}∥_{H^{2}}^{2}.

Since the H^{2} norm of any power of z is 1 and 0 ≤ H ≤ 1, it follows that Hz^{k} = z^{k}. Since every polynomial of degree at most n is a linear combination of monomials of the form z^{k} with z as above and k = 0, 1, …, n, the proof of Theorem 4.9 is complete □ 5. The Toeplitz C^{∗}-algebra. Let S̄ = (S_{1}, …, S_{d}) be the d-shift.

Definition 5.1. The Toeplitz C^{∗}-algebra is the C^{∗}-algebra T_{d} generated by the operators S_{1}, …, S_{d}.

Remarks. Notice that we have not included the identity operator as one of the generators of T_{d}, so that T_{d} is by definition the norm closed linear span of the set of finite products of the form T_{1}T_{2} … T_{n}, n = 1, 2, …, where

T_{i} ∈ {S_{1}, …, S_{d}, S_{1}^{∗}, …, S_{d}^{∗}}.

Nevertheless, (5.5) below implies that T_{d} contains an invertible positive operator

(d1 + N)(1 + N)^{−1} = S_{1}^{∗}S_{1} + ⋯ + S_{d}^{∗}S_{d},

and hence 1 ∈ T_{d}. Thus T_{d} is the C^{∗}-algebra generated by all multiplication operators M_{f} ∈ B(H^{2}), f ∈ P.

If one starts with the Hilbert space H^{2}(∂B_{d}) rather than H^{2} then there is a natural Toeplitz C^{∗}-algebra

T_{∂B_{d}} = C^{∗}{M_{f} : f ∈ P} ⊆ B(H^{2}(∂B_{d})),

and similarly there is a Toeplitz C^{∗}-algebra T_{B_{d}} on the Bergman space

T_{B_{d}} = C^{∗}{M_{f} : f ∈ P} ⊆ B(H^{2}(B_{d}))

[16]. In fact, it is not hard to show that the three C^{∗}-algebras T_{d}, T_{∂B_{d}} and T_{B_{d}} are unitarily equivalent. In that sense, the C^{∗}-algebra T_{d} is not new. However, we are concerned with the relationship between the d-shift and its enveloping C^{∗}-algebra T_{d}, and here there are some essential differences.

For example, in the classical case of H^{2}(∂B_{d}) one can start with a continuous complex-valued function f ∈ C(∂B_{d}) and define a Toeplitz operator T_{f} on H^{2}(∂B_{d}) by compressing the operator of multiplication by f (acting on L^{2}(∂B_{d})) to the subspace H^{2}(∂B_{d}). In our case however, continuous symbols do not give rise to Toeplitz operators. Indeed, we have seen that there are continuous functions f on the closed unit ball which are uniform limits of holomorphic polynomials, but which do not belong to H^{2}. For such an f the “Toeplitz” operator T_{f} is not defined. Thus we have taken some care to develop the properties of T_{d} that we require.

Let N be the number operator acting on H^{2}, defined as the generator of the one-parameter unitary group

Γ(e^{it}1_{E}) = e^{itN}, t ∈ ℝ,

Γ being the representation of the unitary group of E on H^{2} defined in the remarks following Definition 2.10. N obviously has discrete spectrum {0, 1, 2, … } and the nth eigenspace of N is the space P_{n} of homogeneous polynomials of degree n,

P_{n} = {ξ ∈ H^{2} : Nξ = nξ}, n = 0, 1, 2, ….

(1 + N)^{−1} is a compact operator, and it is a fact that for every real number p > 0,

(5.2)  trace (1 + N)^{−p} < ∞ ⟺ p > d.

Since N is unitarily equivalent to the Bosonic number operator, the assertion (5.2) is probably known. We lack an appropriate reference, however, and have included a proof of (5.2) in the Appendix for the reader’s convenience.

The following result exhibits the commutation relations satisfied by the d-shift.

Proposition 5.3. Suppose that d = 2, 3, … and let (S_{1}, …, S_{d}) be the d-shift. Then for all i, j = 1, …, d we have

(5.4)  S_{i}^{∗}S_{j} − S_{j}S_{i}^{∗} = (1 + N)^{−1}(δ_{ij}1 − S_{j}S_{i}^{∗}),

and

(5.5)  S_{1}^{∗}S_{1} + ⋯ + S_{d}^{∗}S_{d} = (d1 + N)(1 + N)^{−1}.

In particular, ∥S_{1}^{∗}S_{1} + ⋯ + S_{d}^{∗}S_{d}∥ = d. The commutators S_{i}^{∗}S_{j} − S_{j}S_{i}^{∗} belong to every Schatten class L^{p}(H^{2}) for p > d, but they do not belong to L^{d}(H^{2}).

Remark. It follows that if A, B are operators belonging to the unital ∗-algebra generated by S_{1}, …, S_{d}, then AB − BA ∈ L^{p}(H^{2}) for every p > d, and hence any product of at least d + 1 such commutators belongs to the trace class.

proof. To establish these formulas it is more convenient to work with the d-shift in its realization on F_{+}(E) described in Proposition 2.13. Thus, we pick an orthonormal basis e_{1}, …, e_{d} for a d-dimensional Hilbert space E and set

S_{i}ξ = e_{i}ξ, 1 ≤ i ≤ d,

for ξ ∈ F_{+}(E) = ℂ ⊕ E ⊕ E^{2} ⊕ . . .. The number operator N acts as follows on E^{n}:

Nξ = nξ, ξ ∈ E^{n}, n = 0, 1, 2, ….

We first establish (5.4). It suffices to verify that the operators on both sides of (5.4) agree on every finite dimensional space E^{n}, n = 0, 1, 2, . . .. For n = 0 and λ ∈ ℂ we have S_{i}^{∗}S_{j}λ = λS_{i}^{∗}e_{j} = δ_{ij}λ, while S_{j}S_{i}^{∗}λ = 0, hence (5.4) holds on ℂ. For n ≥ 1 and ξ ∈ E^{n} of the form ξ = y^{n} we see from formula (3.9) that

S_{j}^{∗}S_{i}ξ = S_{j}^{∗}(e_{i}y^{n}) = \frac{δ_{ij}}{n + 1}y^{n} + \frac{n}{n + 1} ⟨y, e_{j}⟩ e_{i}y^{n−1}

while

S_{i}S_{j}^{∗}ξ = ⟨y, e_{j}⟩ S_{i}y^{n−1} = ⟨y, e_{j}⟩ e_{i}y^{n−1}.

Hence

S_{j}^{∗}S_{i}ξ − S_{i}S_{j}^{∗}ξ = \frac{1}{n + 1}(δ_{ij}ξ − S_{j}S_{i}^{∗}ξ).

The latter holds for all ξ ∈ E^{n} because E^{n} is spanned by {y^{n} : y ∈ E}, and (5.4) follows.

Formula (5.5) follows from (5.4). Indeed, for ξ ∈ E^{n} we have

S_{i}^{∗}S_{i}ξ = \frac{1}{n + 1}ξ + \frac{n}{n + 1}S_{i}S_{i}^{∗}ξ.

By the remarks following (2.10) we have

(5.6)  S_{1}S_{1}^{∗} + ⋯ + S_{d}S_{d}^{∗} = 1 − E_{0},

E_{0} denoting the projection on ℂ. Summing the previous formula on i we obtain

∑_{i=1}^{d} S_{i}^{∗}S_{i}ξ = \frac{d}{n + 1}ξ + \frac{n}{n + 1}(ξ − E_{0}ξ) = \frac{n + d}{n + 1}ξ − \frac{n}{n + 1}E_{0}ξ = \frac{n + d}{n + 1}ξ,

and (5.5) follows.

Now suppose p > d. Because of (5.2) the operator (1 + N)^{−1} belongs to L^{p}; since L^{p} is an ideal (5.4) implies that S_{i}^{∗}S_{j} − S_{j}S_{i}^{∗} ∈ L^{p} for all i, j.

Finally, we claim that no self-commutator [S_{i}^{∗}, S_{i}] = S_{i}^{∗}S_{i} − S_{i}S_{i}^{∗} belongs to L^{d}. Indeed, since the operators S_{1}, …, S_{d} are unitarily equivalent to each other (by the remarks following Definition 2.10), we see that if one [S_{i}^{∗}, S_{i}] belongs to L^{d} then they all do, and in that case we would have

∑_{i=1}^{d}[S_{i}^{∗}, S_{i}] ∈ L^{d}.

By (5.5) and (5.6) the left side of this formula is

∑_{i=1}^{d} S_{i}^{∗}S_{i} − ∑_{i=1}^{d} S_{i}S_{i}^{∗} = (d1 + N)(1 + N)^{−1} − (1 − E_{0})
= E_{0} + (d − 1)(1 + N)^{−1}.

Since (1+N)^{−1} ∉ L^{d} by (5.2), we have a contradiction and the proof of Proposition 5.3 is complete □

The d-shift and the Canonical Commutation Relations. The d-shift is closely related to the creation operators (C_{1}, …, C_{d}) associated with the canonical commutation relations for d degrees of freedom. Indeed, one can think of S̄ = (S_{1}, …, S_{d}) as the partial isometry occurring in the polar decomposition of C̄ = (C_{1}, …, C_{d}) in the following way. Choose an orthonormal basis e_{1}, …, e_{d} for a d-dimensional Hilbert space E. For k = 1, …, d, C_{k} is defined on the dense subspace of F_{+}(E) spanned by E^{n}, n = 0, 1, … as follows

C_{k}ξ = \sqrt{n + 1}e_{k}ξ, ξ ∈ E^{n}

(see [40]). The C_{k} are of course unbounded operators, and they satisfy the complex form of the canonical commutation relations

C_{i}C_{j} = C_{j}C_{i},  C_{i}^{∗}C_{j} − C_{j}C_{i}^{∗} = δ_{ij}1,  1 ≤ i, j ≤ d.

One finds that the row operator

C̄ = (C_{1}, …, C_{d}) : \underbrace{F_{+}(E) ⊕ ⋯ ⊕ F_{+}(E)}_{d times} → F_{+}(E)

is related to the number operator N by C̄C̄^{∗} = N, and in fact the polar decomposition of C̄ takes the form

C̄ = N^{1/2}S̄

where S̄ = (S_{1}, …, S_{d}) is the d-shift; i.e., C_{k} = N^{1/2}S_{k}, k = 1, …, d.

We have seen that the d-shift is not a subnormal d-tuple. The following result asserts that, at least, the individual operators S_{k}, k = 1, …, d are hyponormal. Indeed, any linear combination of S_{1}, …, S_{d} is a hyponormal operator.

Corollary. For every k = 1, …, d we have S_{k}^{∗}S_{k} ≥ S_{k}S_{k}^{∗}.

proof. Proposition 5.3 implies that

S_{k}^{∗}S_{k} − S_{k}S_{k}^{∗} = (1 + N)^{−1}(1 − S_{k}S_{k}^{∗}).

Since ∥S_{k}∥ ≤ 1, both factors on the right are positive operators. Let E_{n} be the nth spectral projection of N, n = 0, 1, . . .. Since S_{k}E_{n} = E_{n+1}S_{k} it follows that S_{k}S_{k}^{∗} commutes with E_{n}. Thus (1 + N)^{−1} commutes with 1 − S_{k}S_{k}^{∗}, and the assertion follows. □

Of course in dimension d = 1, the commutator S^{∗}S −SS^{∗} is a rank-one operator and therefore belongs to every Schatten class L^{p}, p ≥ 1.

Theorem 5.7. T_{d} contains the algebra K of all compact operators on H^{2}, and we have an exact sequence of C^{∗}-algebras

0 ⟶ K ↪ T_{d} ⟶_{π} C(∂B_{d}) ⟶ 0

where π is the unital ∗-homomorphism defined by

π(S_{k}) = z_{k},

z_{k} being the kth coordinate function z_{k}(x) = ⟨x, e_{k}⟩, x ∈ ∂B_{d}.

Letting A be the commutative algebra of polynomials in the operators S_{1}, …, S_{d} we have

(5.8)  T_{d} = \overline{span}AA^{∗}.

proof. Let E_{0} be the one-dimensional projection onto the space of constants in H^{2}. By the remark following Definition 2.10 we have

E_{0} = 1 − S_{1}S_{1}^{∗} − ⋯ − S_{d}S_{d}^{∗} ∈ spanAA^{∗}.

Thus for any polynomials f, g, the the rank-one operator

f ⊗ ḡ : ξ ↦ ⟨ξ, g⟩ f

can be expressed as

f ⊗ ḡ = M_{f}E_{0}M_{g}^{∗} ∈ spanAA^{∗}.

It follows that the norm closure of spanAA^{∗} contains the algebra K of all compact operators.

By Proposition 5.3, the quotient T_{d}/K is a commutative C^{∗}-algebra which is generated by commuting normal elements Z_{k} = π(S_{k}), k = 1, …, d satisfying

Z_{1}Z_{1}^{∗} + ⋯ + Z_{d}Z_{d}^{∗} = 1.

Because T_{d} is commutative modulo K and since \overline{span}AA^{∗} contains K, it follows that \overline{span}AA^{∗} is closed under multiplication and (5.8) follows.

Let X be the joint spectrum of the commutative normal d-tuple (Z_{1}, …, Z_{d}) that generates T_{d}/K. X is a nonvoid subset of the sphere ∂B_{d}, and we claim that X = ∂B_{d}. Indeed, since the unitary group U(E) acts transitively on ∂B_{d} it suffices to show that for every unitary d × d matrix u = (u_{ij}), there is a ∗-automorphism θ_{u} of T_{d}/K such that

θ_{u}(Z_{i}) = ∑_{j=1}^{d} ū_{ji}Z_{j}.

For that, consider the unitary operator U acting on E by

Ue_{i} = ∑_{j=1}^{d} ū_{ji}e_{j}.

Then Γ(U) is a unitary operator on H^{2} for which

Γ(U)S_{i}Γ(U)^{∗} = ∑_{j=1}^{d} ū_{ji}S_{j},

hence θ_{u} is obtained by promoting the spatial automorphism T ↦ Γ(U)T Γ(U)^{∗} of T_{d} to the quotient T_{d}/K.

The identification of T_{d}/K with C(∂B_{d}) asserted by π(S_{i}) = z_{i}, i = 1, …, d is now obvious □

6. d-contractions and A-morphisms.

The purpose of this section is to make some observations about the role of Amorphisms in function theory and operator theory.

Definition 6.1. Let A be a subalgebra of a unital C^{∗}-algebra B which contains the unit of B. An A-morphism is a completely positive linear map ϕ : B → B(H) of B into the operators on a Hilbert space H such that ϕ(1) = 1 and

ϕ(AX) = ϕ(A)ϕ(X) A ∈ A, X ∈ B.

A-morphisms arose naturally in our work on the dilation theory of completely positive maps and semigroups [7,8,9]. Jim Agler has pointed out that they are related to his notion of hereditary polynomials and hereditary isomorphisms (for example, see [1, Theorem 1.5]). Indeed, if B denotes the C^{∗}-algebra generated by a single operator T and the identity, then one can show that a completely positive map of B which is a hereditary isomorphism on the space of hereditary polynomials in T is an A-morphism relative to the algebra A of all polynomials in the adjoint T^{∗}.

In general the restriction of an A-morphism to A is a completely contractive representation of the subalgebra A on H. Theorem 4.5 implies that every d-contraction T̄ acting on a Hilbert space H gives rise to a contraction L : F_{+}(ℂ^{d}) ⊗ K → H which intertwines the action of the d-shift and T̄. L is often a co-isometry, and that implies the following assertion about A-morphisms.

Theorem 6.2. Let A be the subalgebra of the Toeplitz C^{∗}-algebra T_{d} consisting of all polynomials in the d-shift (S_{1}, …, S_{d}). Then for every d-contraction (T_{1}, …, T_{d}) acting on a Hilbert space H there is a unique A-morphism

ϕ : T_{d} → B(H)

such that ϕ(S_{k}) = T_{k}, k = 1, …, d.

Conversely, every A-morphism ϕ : T_{d} → B(H) gives rise to a d-contraction (T_{1}, …, T_{d}) on H by way of T_{k} = ϕ(S_{k}), k = 1, …, d.

proof. The uniqueness assertion is immediate from (5.8), since an A-morphism is uniquely determined on the closed linear span of the set of products {AB^{∗} : A, B ∈ A}.

For existence, we first show that every null d-contraction T̄ = (T_{1}, …, T_{d}) defines an A-morphism as asserted in Theorem 6.2. For that, let

Δ = (1 − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗})^{1/2},

let K = \overline{ΔH} be the closed range of Δ and let F_{+}(E) be the symmetric Fock space over E = ℂ^{d}. Choose an orthonormal basis e_{1}, …, e_{d} for E. Theorem 4.5 asserts that there is a unique bounded operator L : F_{+}(E) ⊗ K → H satisfying L(1 ⊗ ξ) = Δξ for ξ ∈ K and

(6.3)  L(e_{i_{1}}e_{i_{2}} … e_{i_{n}} ⊗ ξ) = T_{i_{1}}T_{i_{2}} … T_{i_{n}}Δξ,

for n = 1, 2, …, i_{1}, i_{2}, …, i_{n} ∈ {1, …, d}, ξ ∈ K; moreover, since (T_{1}, …, T_{d}) is a null d-contraction L is a coisometry.

We may consider that the d-shift (S_{1}, …, S_{d}) is defined on F_{+}(E) by

S_{k}ξ = e_{k}ξ, k = 1, …, d.

(6.3) implies that

(6.4)  L(f(S_{1}, …, S_{d}) ⊗ 1_{K}) = f(T_{1}, …, T_{d})L

for every polynomial f in d variables. Let ϕ : T_{d} → B(H) be the completely positive map

ϕ(X) = L(X ⊗ 1_{K})L^{∗}, X ∈ T_{d}.

Since L^{∗} is an isometry we have ϕ(1) = 1_{H}. (6.4) implies that for every X ∈ T_{d} we have

ϕ(f(S_{1}, …, S_{d})X) = f(T_{1}, …, T_{d})ϕ(X),

hence ϕ is an A-morphism having the required properties.

The general case is deduced from this by a simple device. Let T̄ = (T_{1}, …, T_{d}) be any d-contraction, choose a number r so that 0 < r < 1, and set

T̄_{r} = (rT_{1}, …, rT_{d}).

The row norm of the d-tuple T̄_{r} is at most r, hence T̄_{r} is a null d-contraction (see Remark 4.4). By what was just proved there is an A-morphism ϕ_{r} : T_{d} → B(H) satisfying

ϕ_{r}(S_{k}) = rT_{k}, k = 1, …, d.

We have

ϕ_{r}(f(S_{1}, …, S_{d})g(S_{1}, …, S_{d})^{∗}) = f(rT_{1}, …, rT_{d})g(rT_{1}, …, rT_{d})^{∗}

for polynomials f, g. Since operators of the form f(S_{1}, …, S_{d})g(S_{1}, …, S_{d})^{∗} span T_{d} and since the family of maps ϕ_{r}, 0 < r < 1 is uniformly bounded, it follows that ϕ_{r} converges point-norm to an A-morphism ϕ as r ↑ 1, and ϕ(S_{k}) = T_{k} for all k.

It remains only to show that for every A-morphism ϕ : T_{d} → B(H), the operators T_{k} = ϕ(S_{k}) define a d-contraction. To see that, write

T_{k}T_{k}^{∗} = ϕ(S_{k})ϕ(S_{k})^{∗} = ϕ(S_{k}S_{k}^{∗}).

Then

∑_{k=1}^{d} T_{k}T_{k}^{∗} = ϕ(∑_{k=1}^{d} S_{k}S_{k}^{∗}) ≤ ϕ(1) = 1.

So by remark 3.2, (T_{1}, …, T_{d}) is a d-contraction □

Remarks. We have already pointed out that in general, an A-morphism must be a completely contractive representation of A. Conversely, if A is the polynomial algebra in T_{d} and ϕ : A → B(H) is a representation which is d-contractive in the sense that its natural promotion to d × d matrices over A is a contraction, then after noting that the operator matrix

A = ( \begin{matrix}S_{1} & S_{2} & … & S_{d} \\ 0 & 0 & … & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & 0 \end{matrix} ) ∈ M_{d}(T_{d})

satisfies ∥A∥^{2} = ∥AA^{∗}∥ = ∥S_{1}S_{1}^{∗} + ⋯ + S_{d}S_{d}^{∗}∥ = 1, we find that the image of A under the promotion of ϕ is a contraction, and hence T_{k} = ϕ(S_{k}), k = 1, …, d defines a d-contraction. Thus, we may conclude

Corollary 1. Let d = 1, 2, . . .. Every d-contractive representation ϕ of the polynomial algebra A ⊆ T_{d} is completely contractive, and can be extended uniquely to an A-morphism

ϕ̃: T_{d} → B(H).

We have already seen that the unitary group U_{d} of ℂ^{d} acts naturally on T_{d} as a group of ∗-automorhisms by way of

θ_{U}(X) = Γ(U)XΓ(U)^{∗}, X ∈ T_{d}, U ∈ U_{d}.

As a straightforward application of Theorem 6.2 we show that the definition of θ can be extended to all contractions in B(ℂ^{d}) so as to obtain a semigroup of A-morphisms acting on T_{d}.

Corollary 2. Let A ⊆ T_{d} be the algebra of all polynomials in S_{1}, …, S_{d}. For every contraction A acting on ℂ^{d} there is a unique A-morphism θ_{A} : T_{d} → B(H^{2}) satisfying

(6.5)  θ_{A}(M_{f}) = M_{f∘A^{∗}}

for every linear functional f on ℂ^{d}, A^{∗} denoting the adjoint of A ∈ B(ℂ^{d}).

proof. Considering the polar decomposition of A, we may find a pair of orthonormal bases u_{1}, …, u_{d} and u′_{1}, …, u′_{d} for ℂ^{d} and numbers λ_{k} in the unit interval such that

Au_{k} = λ_{k}u′_{k}, k = 1, …, d.

Let z_{1}, …, z_{d} and z′_{1}, …, z′_{d} be the corresponding systems of orthogonal coordinate functions

z_{k}(x) = ⟨x, u_{k}⟩ ,
z′_{k}(x) = ⟨x, u′_{k}⟩ .

The linear functionals z_{k}, z′_{k} are related by

(6.6)  z_{k} ∘ A^{∗} = λ_{k}z′_{k},  k = 1, …, d.

Thus if we realize the d-shift (S_{1}, …, S_{d}) as S_{k} = M_{z_{k}} and if we set T_{k} = λ_{k}M_{z′_{k}}, then (T_{1}, …, T_{d}) is a d-contraction and Theorem 6.2 implies that there is a unique A-morphism θ_{A} : T_{d} → B(H^{2}) such that θ_{A}(S_{k}) = T_{k} for every k. After noting that θ_{A} satisfies (6.5) because of (6.6) above, the proof is complete □

From (6.5) together with the uniqueness assertion of Theorem 6.2 it follows that for two contractions A, B ∈ B(ℂ^{d}) we have θ_{AB} = θ_{A} ∘ θ_{B}. It is routine to verify that θ_{A}(T_{d}) ⊆ T_{d}, that for every fixed X ∈ T_{d} the function A ↦ θ_{A}(X) moves continuously in the norm of T_{d}, and that θ_{A} agrees with the previous definition when A is unitary.

Uniqueness of Representing measures. Representing measures for points in the interior of the unit ball in ℂ^{d} are notoriously non-unique in dimension d ≥ 2. Indeed, for every t̄ = (t_{1}, …, t_{d}) ∈ B_{d} there is an uncountable family of probability measures μ_{α} supported in the boundary ∂B_{d} such that μ_{α} ⊥ μ_{β} for α ≠ β and

∫_{∂B_{d}} f(ζ) dμ_{α}(ζ) = f(t̄), f ∈ P

see [37, p. 186]. The following result asserts that one can recover uniqueness by replacing measures on ∂B_{d} with states on the Toeplitz C^{∗}-algebra which define A-morphisms.

Corollary 3. Let t̄ = (t_{1}, …, t_{d}) ∈ ℂ^{d} satisfy |t_{1}|^{2} + ⋯ + |t_{d}|^{2} < 1 and let S̄ = (S_{1}, …, S_{d}) be the d-shift. Then there is a unique state ϕ of T_{d} satisfying

(6.7)  ϕ(f(S̄)g(S̄)^{∗}) = f(t̄)ḡ(t̄),  f, g ∈ P.

ϕ is the (pure) vector state

ϕ(A) = (1 − ∥t̄∥^{2}) ⟨Aut̄, ut̄⟩ , A ∈ T_{d}

where ut̄(x) = (1 − ⟨x, t̄⟩)^{−1} is the H^{2} function defined in (1.11).

proof. We may consider that t̄ = (t_{1}, …, t_{d}) is a d-contraction acting on the onedimensional Hilbert space ℂ. Theorem 6.2 implies that there is a unique state ϕ : T_{d} → ℂ satisfying (6.7), and it remains only to identify ϕ. From (2.4) we have

⟨M_{f}M_{g}^{∗}ut̄, ut̄⟩ = ⟨M_{g}^{∗}ut̄, M_{f}^{∗}ut̄⟩ = f(t̄)ḡ(t̄)∥ut̄∥^{2} = (1 − ∥t̄∥^{2})^{−1}f(t̄)ḡ(t̄)

as asserted. □

7. The d-shift as an operator space.

In this section we consider the operator space S_{d} ⊆ B(H^{2}) generated by the d-shift (S_{1}, …, S_{d}),

S_{d} = {a_{1}S_{1} + ⋯ + a_{d}S_{d} : a_{1}, …, a_{d} ∈ ℂ}.

By a commutative operator space we mean a linear subspace S ⊆ B(H) whose operators mutually commute with one another. We introduce a sequence of numerical invariants for arbitrary operator spaces, and for dimension d ≥ 2 we show that among all d-dimensional commutative operator spaces, S_{d} is distinguished by the fact that its sequence of numerical invariants is maximal (Theorem 7.7).

Given an arbitrary operator space S ⊆ B(H), let T̄ = (T_{1}, T_{2}, …) be an infinite sequence of operators in S such that all but a finite number of terms are 0. We write seq(S) for the set of all such sequences. Every such sequence has a “row norm” and a “column norm”, depending on whether one thinks of the sequence as defining an operator in B(H^{∞}, H) or in B(H, H^{∞}). These two norms are familiar and easily computed,

∥T̄∥_{row} = ∥ ∑_{k} T_{k}T_{k}^{∗}∥^{1/2}
∥T̄∥_{col} = ∥ ∑_{k} T_{k}^{∗}T_{k}∥^{1/2}.

Given two sequences T̄, T̄′ ∈ seq(S), we can form a product sequence (T_{i}T′_{j} : i, j = 1, 2, …) which we may consider an element of seq(B(H)), if we wish, by relabelling the double sequence as a single sequence. Though for the computations below it will be more convenient to allow the index set to vary in the obvious way. In particular, every T̄ ∈ seq(S) can be raised to the nth power to obtain T̄^{n} ∈ seq(B(H)), n = 1, 2, . . .. For each n = 1, 2, … we define E_{n}(S) ∈ [0, +∞] as follows,

E_{n}(S) = sup{∥T̄^{n}∥_{col}^{2} : T̄ ∈ seq(S), ∥T̄∥_{row} ≤ 1}.

In the most explicit terms, we have

E_{n}(S) = sup{∥ ∑_{i_{1},...,i_{n}=1}^{∞} T_{i_{1}}^{∗} … T_{i_{n}}^{∗}T_{i_{n}} … T_{i_{1}}∥ : T_{i} ∈ S, ∥ ∑_{i=1}^{∞} T_{i}T_{i}^{∗}∥ ≤ 1},

the sup being taken over finitely nonzero sequences T_{i} ∈ S.

Definition 7.1. E_{1}(S), E_{2}(S), … is called the energy sequence of the operator space S.

If S is the one-dimensional space spanned by a single operator T of norm 1, then the energy sequence degenerates to E_{n}(S) = ∥T^{n}∥^{2}, n = 1, 2, . . .. In general, E_{n}(S)^{1/2} is the norm of the homogeneous polynomial T̄ ↦ T̄^{n}, considered as a map of row sequences in S to column sequences in B(H).

Remarks. We have defined the energy sequence in elementary terms. It is useful, however, to relate it to completely positive maps. Fixing an operator space S, notice that every sequence T̄ ∈ seq(S) gives rise to a normal completely positive map PT̄ on B(H) as the sum of the finite series

(7.2)  PT̄(A) = T_{1}AT_{1}^{∗} + T_{2}AT_{2}^{∗} + ….

Let cp(S) denote the set of all completely positive maps of the form (7.2). The norm of P = PT̄ is given by

∥P ∥ = ∥P (1)∥ = ∥T̄∥_{row}.

Now any map P ∈ cp(S) of the form (7.2) has an adjoint P_{∗} which is defined as the completely positive map satisfying

trace (P (A)B) = trace (AP_{∗}(B))

for all finite rank operators A, B. One finds that if P ∈ cp(S) is given by the finitely nonzero sequence T̄ then P_{∗} ∈ cp(S^{∗}) is given by the sequence of adjoints

(7.3)  P_{∗}(A) = T_{1}^{∗}AT_{1} + T_{2}^{∗}AT_{2} + ….

Of course P, being a normal linear map of B(H), is the adjoint of a bounded linear map P_{∗} acting on the predual of B(H), and the map of (7.3) is simply this preadjoint extended from the trace class operators to all of B(H) (note that we use the fact that the sequence T̄ is finitely nonzero here, since in general a bounded linear map of the trace class operators can be unbounded relative to the operator norm, and thus not extendable up to B(H)).

In any case, we find that if P ∈ cp(S) has the form P = PT̄ for T̄ ∈ seq(S) then

∥P_{∗}∥ = ∥P_{∗}(1)∥ = ∥T̄∥_{col}.

Thus the definition of E_{n}(S) can be restated as follows,

(7.4)  E_{n}(S) = sup{∥P_{∗}^{n}∥ : P ∈ cp(S), ∥P ∥ ≤ 1}.

The following result implies that for a finite dimensional operator space S the terms of the energy sequence are all finite, and if S is commutative then they grow no faster than E_{n}(S) = O(n^{d−1}), where d is the dimension of S.

Proposition 7.5. Let S be an operator space of finite dimension d. Then

E_{n}(S) ≤ d^{n},

and if S is also commutative then

E_{n}(S) ≤ \frac{d(d + 1) … (d + n − 1)}{n!} = \frac{(n + d − 1)!}{n!(d − 1)!}.

proof. Let P ∈ cp(S) satisfy ∥P ∥ ≤ 1, and let d be the dimension of S. It is clear that the metric operator space [8] of P is a subspace of S, and in particular there is a linearly independent set of r ≤ d elements T_{1}, …, T_{r} in S such that

P (A) = T_{1}AT_{1}^{∗} + … T_{r}AT_{r}^{∗}, A ∈ B(H).

Since ∥P ∥ = ∥P (1)∥ = ∥T_{1}T_{1}^{∗} + ⋯ + T_{r}T_{r}^{∗}∥ ≤ 1 it follows that ∥T_{k}∥ ≤ 1 for every k = 1, …, r, hence

∥P_{∗}∥ = ∥T_{1}^{∗}T_{1} + ⋯ + T_{r}^{∗}T_{r}∥ ≤ r ≤ d.

Thus ∥P_{∗}^{n}∥ ≤ d^{n} for every n = 1, 2, . . .. From (7.4) we conclude that E_{n}(S) ≤ d^{n}.

In fact, the preceding argument shows that if Q is a normal completely positive map of B(H) whose metric operator space is r-dimensional and which satisfies ∥Q∥ ≤ 1, then we have ∥Q_{∗}∥ ≤ r.

We apply this to Q = P^{n} as follows. By [8], the metric operator space E_{n} of P^{n} is a subspace of

span{L_{1}L_{2} … L_{n} : L_{i} ∈ S}.

Assuming S to be commutative, the latter is naturally isomorphic to a quotient of the n-fold symmetric tensor product of vector spaces S^{n}. Since

dim S^{n} = \frac{(n + d − 1)!}{n!(d − 1)!}

(see formula A.5 of the Appendix), and since ∥P^{n}∥ ≤ 1, we find that

∥P_{∗}^{n}∥ = ∥(P^{n})_{∗}∥ ≤ dim E_{n} ≤ \frac{(n + d − 1)!}{n!(d − 1)!}.

The required estimate follows from the observation (7.4). □

Remark. The asserted growth rate of the binomial coefficients of Proposition 7.5 is well known, and the precise asymptotic relation is reiterated in formula A.6.

Throughout the remainder of this section we will be concerned with finite dimensional commutative operator spaces.

Definition 7.6. A commutative operator space S of finite dimension d is said to be maximal if for every n = 1, 2, … we have

E_{n}(S) = \frac{(n + d − 1)!}{n!(d − 1)!}.

Remarks. It is obvious that if S is a space of mutually commuting normal operators, then E_{n}(S) = 1 for every n. Similarly, it is not hard to show that if S is a finite dimensional space of commuting quasinilpotent operators, then

lim_{n→∞} E_{n}(S) = 0.

Thus the maximal spaces are rather far removed from both of these types.

It is also true (though less obvious) that if S is a commutative operator space of dimension d for which

E_{n}(S) = \frac{(n + d − 1)!}{n!(d − 1)!}

for some particular value of n ≥ 2, then

E_{k}(S) = \frac{(k + d − 1)!}{k!(d − 1)!}

for every k = 1, 2, …, n. Thus for operator spaces which are not maximal, once the sequence of numbers E_{n}(S) departs from the sequence of maximum possible values, it never returns. We omit the proof of these assertions since they are not required in the sequel.

Theorem 7.7. For every d = 1, 2, … the operator space S_{d} of the d-shift is maximal.

Conversely, if d ≥ 2 and if S is a d-dimensional commutative operator space which is maximal, then there is a representation π of the unital C^{∗}-algebra C^{∗}(S) generated by S on H^{2} such that π(S) = S_{d}. In particular, the Toeplitz C^{∗}-algebra T_{d} is isomorphic to a quotient of C^{∗}(S).

Before giving the proof of 7.7, we deduce from it the following characterization of S_{d} as a space of essentially normal operators (by that we mean a commuting family of operators in B(H) whose image in the Calkin algebra consists of normal elements). We remark that both the Corollary and the essential part of Theorem 7.7 are false in dimension one.

Corollary. Assume d ≥ 2. Up to unitary equivalence, the space S_{d} spanned by the d-shift is the only d-dimensional irreducible commutative operator space consisting of essentially normal operators, which is maximal in the sense of Definition 7.6.

proof of Corollary. Suppose that S acts on a Hilbert space H, and let K denote the algebra of all compact operators on H. Let π : C^{∗}(S) → B(H^{2}) be the representation of Theorem 7.7. The operators in S cannot be normal because S_{d} = π(S) contains no normal operators. Since [S^{∗}, S] ⊆ K ∩ C^{∗}(S) and since C^{∗}(S) is irreducible, it follows that C^{∗}(S) contains K. π(K) cannot be {0} because that would imply that π(S) = S_{d} consists of normal operators.

Thus π is an irreducible representation of C^{∗}(S) which is nonzero on K, hence π must be unitarily equivalent to the identity representation of C^{∗}(S). In particular, S is unitarily equivalent to S_{d} □

proof of Theorem 7.7. The proof of Theorem 7.7 will occupy the remainder of this section. Let (S_{1}, …, S_{d}) be the d-shift, let S_{d} = span{S_{1}, …, S_{d}} be its associated operator space, and define P ∈ cp(S_{d}) by

P (A) = S_{1}AS_{1}^{∗} + ⋯ + S_{d}AS_{d}^{∗}.

By the remark following Definition 2.10 we have P (1) = 1 − E_{0}, hence ∥P ∥ = 1. Thus to show that S_{d} is maximal it suffices to show that for each n ≥ 1, the operator P_{∗}^{n}(1) satisfies

(7.8)  ∥P_{∗}^{n}∥ = ∥P_{∗}^{n}(1)∥ = \frac{(n + d − 1)!}{n!(d − 1)!}.

While (7.8) can be deduced directly from Corollary 4.9, we actually require somewhat more information about the operators P_{∗}^{n}(1) and their eigenvalue distributions.

Lemma 7.9. Let N = E_{1} + 2E_{2} + 3E_{3} + … be the number operator acting on H^{2} and for every n = 1, 2, … let g_{n} : [0, ∞) → ℝ be the bounded continuous function

g_{n}(x) = ∏_{k=1}^{n} \frac{x + k + d − 1}{x + k}.

Then

P_{∗}^{n}(1) = g_{n}(N) = ∑_{k=0}^{∞} g_{n}(k)E_{k}.

The eigenvalue sequence {g_{n}(0) ≥ g_{n}(1) ≥ … } of P_{∗}^{n}(1) is decreasing and we have

∥P_{∗}^{n}(1)∥ = g_{n}(0) = \frac{(n + d − 1)!}{n!(d − 1)!}.

If d ≥ 2 then the eigenvalue sequence is strictly decreasing, g_{n}(0) > g_{n}(1) > . . ..

proof of Lemma 7.9. The assertions follow from a direct computation, which can be organized as follows. By Proposition 5.3 we have

(7.10)  P_{∗}(1) = g_{1}(N),

where N is the number operator and g_{1} is the function of a real variable defined by

g_{1}(x) = \frac{x + d}{x + 1}, x ≥ 0.

More generally, if g is any bounded continuous function defined on [0, ∞), then we have

(7.11)  P_{∗}(g(N)) = g̃(N)

where

g̃(x) = g(x + 1)\frac{x + d}{x + 1}, x ≥ 0.

Indeed, (7.11) follows from the fact that if E_{k} denotes the kth spectral projection of N,

N = ∑_{k=1}^{∞} kE_{k},

then E_{k} is the projection on the subspace of homogeneous polynomials of degree k in H^{2}, and thus for each i = 1, …, d we have the commutation formulas S_{i}^{∗}E_{0} = 0, and S_{i}^{∗}E_{k} = E_{k−1}S_{i}^{∗} for k ≥ 1. It follows that P_{∗}(E_{0}) = 0 and P_{∗}(E_{k}) = E_{k−1}P_{∗}(1) for k = 1, 2, …, thus

P_{∗}(g(N)) = ∑_{k=1}^{∞} g(k)P_{∗}(E_{k}) = g̃(N).

After iterating (7.11) we find that P_{∗}^{n}(1) = g_{n}(N) where

g_{n}(x) = g_{1}(x)g_{1}(x + 1) … g_{1}(x + n − 1) = ∏_{k=1}^{n} \frac{x + k + d − 1}{x + k}.

Since each g_{n} is a monotone decreasing function we conclude that

∥P_{∗}^{n}(1)∥ = g_{n}(0) = \frac{d(d + 1) … (d + n − 1)}{n!},

and (7.8) follows. It is clear from the recurrence formula for g_{n+1} in terms of g_{n} that when d ≥ 2, g_{n}(x) is a strictly decreasing function of x □ Corollary. Let ω be a state of the Toeplitz algebra T_{d}, d ≥ 2, such that for some n ≥ 1 we have

ω(P_{∗}^{n}(1)) = ∥P_{∗}^{n}(1)∥ = \frac{(n + d − 1)!}{n!(d − 1)!}.

Then ω is the ground state ω(X) = ⟨Xv, v⟩, v denoting the constant function v = 1.

proof. Fix n. By Lemma 7.9 we have

P_{∗}^{n}(1) = λ_{0}E_{0} + λ_{1}E_{1} + …

where λ_{0} > λ_{1} > ⋯ > 0 and λ_{0} = ∥P_{∗}^{n}(1)∥. Thus P_{∗}^{n}(1) has the form

P_{∗}^{n}(1) = λ_{0}(E_{0} + K),

where K is a positive operator satisfying K = (1 − E_{0})K(1 − E_{0}) and ∥K∥ = λ_{1}/λ_{0} < 1. Since ω(P_{∗}^{n}(1)) = λ_{0} we have

ω(E_{0}) + ω(K) = 1.

If ω(E_{0}) < 1 then we would have

ω(K) ≤ ∥K∥ω(1 − E_{0}) = ∥K∥(1 − ω(E_{0})) < 1 − ω(E_{0}),

contradicting the preceding equation. Hence ω(E_{0}) = 1 and ω must be the ground state □

In particular, Lemma 7.9 implies that S_{d} is maximal among all d-dimensional commutative operator spaces.

In order to prove the converse assertion of Theorem 7.7, we recall one or two facts from the theory of boundary representations (see [3], 2.1.2 and 2.2.2). By a unital operator space we mean a pair S ⊆ B consisting of a linear subspace S of a unital C^{∗}-algebra B, which contains the unit of B and generates B as a C^{∗}-algebra, B = C^{∗}(S). An irreducible representation π : B → B(H) is said to be a boundary representation for S if π ↾_{S} has a unique completely positive linear extension to B, namely π itself. Boundary representations are the noncommutative counterpart of points in the Choquet boundary of a function space S ⊆ C(X). Their key property is their functoriality; if S_{1} ⊆ B_{1} and S_{2} ⊆ B_{2} are unital operator spaces and ϕ : S_{1} → S_{2} is a completely isometric linear map satisfying ϕ(1) = 1 and ϕ(S_{1}) = S_{2}, then for every boundary representation π_{2} : B_{2} → B(H) for S_{2} there is a unique boundary representation π_{1} : B_{1} → B(H) for S_{1} which satisfies

(7.12)  π_{2}(ϕ(T)) = π_{1}(T),  T ∈ S_{1}.

Lemma 7.13. For d ≥ 2, the identity representation of the Toeplitz algebra T_{d} is a boundary representation for the d + 1-dimensional space span{1, S_{1}, …, S_{d}}.

proof. By [4, Theorem 2.1.1] it is enough to show that the Calkin map is not isometric when promoted to the space M_{d} ⊗ S of d × d matrices over S. Consider the operator A ∈ M_{d} ⊗ S defined by

A = ( \begin{matrix}S_{1} & 0 & … & 0 \\ S_{2} & 0 & … & 0 \\ ⋮ & ⋮ & & ⋮ \\ S_{d} & 0 & … & 0 \end{matrix} )

Then ∥A^{∗}A∥ = ∥S_{1}^{∗}S_{1} + ⋯ + S_{d}^{∗}S_{d}∥ = d by Proposition 5.3. Hence ∥A∥ = \sqrt{d}. On the other hand, by Theorem 5.7 the Calkin map carries S_{k} to the kth coordinate function z_{k}(x) = ⟨x, e_{k}⟩, x ∈ ∂B_{d}. Hence the image of A under the promoted Calkin map is the matrix of functions on ∂B_{d} defined by

F (x) = ( \begin{matrix}z_{1}(x) & 0 & … & 0 \\ z_{2}(x) & 0 & … & 0 \\ ⋮ & ⋮ & & ⋮ \\ z_{d}(x) & 0 & … & 0 \end{matrix} ) .

Clearly sup\{∥F (x)∥ : x ∈ ∂B_{d}\} = 1 < \sqrt{d} = ∥A∥, as required. □

Lemma 7.13 implies that in dimension d ≥ 2, the d-shift S̄ = (S_{1}, …, S_{d}) can be dilated to another d-contraction T̄ in only a trivial way as a direct summand T_{k} = S_{k} ⊕ Z_{k}, where (Z_{1}, …, Z_{d}) is some d-contraction.

Lemma 7.14. Suppose d ≥ 2, let (T_{1}, …, T_{d}) be a d-contraction acting on a Hilbert space H, and let K ⊆ H be a subspace of H such that the compressed d-tuple

(P_{K}T_{1} ↾_{K}, …, P_{K}T_{d} ↾_{K})

in unitarily equivalent to the d-shift. Then P_{K} commutes with {T_{1}, …, T_{d}}.

proof. By hypothesis, there is an isometry U : H_{d}^{2} → H such that UH_{d}^{2} = K and U^{∗}T_{k}U = S_{k}, k = 1, …, d. By Theorem 6.2, there is an A-morphism ϕ : T_{d} → B(H) such that ϕ(S_{k}) = T_{k}, k = 1, …, d. Define a completely positive map ψ : T_{d} → B(H_{d}^{2}) by ψ(X) = U^{∗}ϕ(X)U. We have ψ(1) = 1, and ψ(S_{k}) = U^{∗}T_{k}U = S_{k}, k = 1, …, d. Lemma 7.13 implies that ψ must be the identity map of T_{d}.

In particular, ψ(XY) = ψ(X)ψ(Y) for all X, Y ∈ T_{d}. Multiplying the latter equation on left and right by U and U^{∗} respectively, we obtain

(7.15)  P_{K}ϕ(XY)P_{K} = P_{K}ϕ(X)P_{K}ϕ(Y)P_{K},  X, Y ∈ T_{d}.

Taking X = Y^{∗} in (7.15) and making use of the Schwarz inequality for completely positive maps we have

((1 − P_{K})ϕ(Y)P_{K})^{∗}(1 − P_{K})ϕ(Y)P_{K} = P_{K}ϕ(Y)^{∗}ϕ(Y)P_{K} − P_{K}ϕ(Y)^{∗}P_{K}ϕ(Y)P_{K}
≤ P_{K}ϕ(Y^{∗}Y)P_{K} − P_{K}ϕ(Y^{∗})P_{K}ϕ(Y)P_{K} = 0

and hence (1 − P_{K})ϕ(Y)P_{K} = 0. Thus K is an invariant subspace for the self adjoint family of operators ϕ(T_{d}), hence P_{K} ∈ ϕ(T_{d})′ ⊆ {T_{1}, …, T_{d}}′. □

Lemma 7.16. Let S ⊆ B(H) be a commutative operator space of finite dimension d ≥ 2, and suppose S is maximal. Then there is a state ρ of the unital C^{∗}-algebra C^{∗}(S) generated by S and a d-contraction T̄ = (T_{1}, …, T_{d}), T_{i} ∈ S, such that

ρ(g(T̄)^{∗}f(T̄)) = ⟨f, g⟩_{H^{2}} ,

for all polynomials f, g ∈ P.

proof of Lemma 7.16. The set of all d-contractions (T_{1}, …, T_{d}) whose component operators belong to S can be regarded as a compact subset of the cartesian product of d copies of the unit ball of S, and of course the state space of C^{∗}(S) is weak^{∗}- compact. Thus, after a routine compactness argument (which we omit), the proof of Lemma 7.16 reduces to establishing the following assertion: for every n = 1, 2, … there is a pair (ρ, T̄) consisting of a state ρ of C^{∗}(S) and a d-contraction T̄ = (T_{1}, …, T_{d}) whose components belong to S such that

(7.17)  ρ(g(T̄)^{∗}f(T̄)) = ⟨f, g⟩_{H^{2}}

for all polynomials f, g ∈ P of degree ≤ n.

To prove the latter, since E_{n}(S) = \frac{(n+d−1)!}{n!(d−1)!} we may find a completely positive map P ∈ cp(S) such that ∥P ∥ ≤ 1 and

(7.18)  ∥P_{∗}^{n}(1)∥ = \frac{(n + d − 1)!}{n!(d − 1)!}

(note that the supremum of (7.4) is achieved here because the space {P ∈ cp(S) : ∥P ∥ ≤ 1} is compact). Considering that the metric operator space of P is a subspace of S [8] we can find a (linearly independent) set T_{1}, …, T_{r} ∈ S such that

P (A) = T_{1}AT_{1}^{∗} + … T_{r}AT_{r}^{∗}, A ∈ B(H).

By appending T_{r+1} = ⋯ = T_{d} = 0 to the sequence if necessary, we can assume that r = d. Because

∥P ∥ = ∥P (1)∥ = ∥T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗}∥ ≤ 1,

T̄ = (T_{1}, …, T_{d}) is a d-contraction for which (7.18) holds.

Let ρ be any state of C^{∗}(S) satisfying

ρ(P_{∗}^{n}(1)) = ∥P_{∗}^{n}(1)∥ = \frac{(n + d − 1)!}{n!(d − 1)!},

and consider the positive semidefinite inner product defined on P by

(7.19)  ⟨f, g⟩ = ρ(g(T̄)^{∗}f(T̄)).

One sees (after consideration of the GNS construction for the state ρ) that since T̄ is a d-contraction, the Hilbert seminorm ∥f∥^{2} = ρ(f(T̄)^{∗}f(T̄)) satisfies

∥z_{1}f_{1} + ⋯ + z_{d}f_{d}∥^{2} ≤ ∥f_{1}∥^{2} + ⋯ + ∥f_{d}∥^{2}

for all polynomials f_{1}, …, f_{d} ∈ P. By Proposition 4.2, ∥ · ∥ will be a contractive seminorm provided that

1 ⊥ z_{1}P + ⋯ + z_{d}P

in its associated inner product space; or equivalently, that

ρ(T_{k}f(T̄)) = 0, k = 1, …, d, f ∈ P.

Since ρ is a state, the latter will follow if we establish

(7.20)  ρ(T_{k}T_{k}^{∗}) = 0,  k = 1, …, d.

To prove 7.20, let ϕ : T_{d} → B(H) be an A-morhpism satisfying ϕ(S_{k}) = T_{k}, k = 1, …, d (see Theorem 6.2), and let ω be the state of T_{d} defined by ω = ρ ∘ ϕ. We claim that ω is the ground state of T_{d}. Indeed, for every n-tuple of integers 1 ≤ i_{1}, …, i_{n} ≤ d we have by the Schwarz inequality

ϕ(S_{i_{1}}^{∗} … S_{i_{n}}^{∗}S_{i_{n}} … S_{i_{1}}) ≥ ϕ(S_{i_{n}} … S_{i_{1}})^{∗}ϕ(S_{i_{n}} … S_{i_{1}}) = T_{i_{1}}^{∗} … T_{i_{n}}^{∗}T_{i_{n}} … T_{i_{1}},

and hence

ω(S_{i_{1}}^{∗} … S_{i_{n}}^{∗}S_{i_{n}} … S_{i_{1}}) ≥ ρ(T_{i_{1}}^{∗} … T_{i_{n}}^{∗}T_{i_{n}} … T_{i_{1}}).

Summing over all such n-tuples we obtain

∑_{i_{1},...,i_{n}=1}^{d} ω(S_{i_{1}}^{∗} … S_{i_{n}}^{∗}S_{i_{n}} … S_{i_{1}}) ≥ ρ(P_{∗}^{n}(1)) = \frac{(n + d − 1)!}{n!(d − 1)!}.

The Corollary of Lemma 7.9 implies that ω must be the ground state of T_{d}. In particular, for each k = 1, …, d we have

ρ(T_{k}T_{k}^{∗}) = ρ(ϕ(S_{k})ϕ(S_{k})^{∗}) = ρ(ϕ(S_{k}S_{k}^{∗})) = ω(S_{k}S_{k}^{∗}) = ∥S_{k}^{∗}1∥^{2} = 0,

and (7.20) follows.

It is clear that the Hilbert seminorm of (7.19) is normalized so that ∥1∥^{2} = ρ(1) = 1, so by Theorem 4.3 we have ∥f∥ ≤ ∥f∥_{H^{2}} for every f ∈ P. Corollary 4.9 now implies that (7.17) is satisfied, and the proof is complete. □

To complete the proof of Theorem 7.7, we find a d-contraction T̄ = (T_{1}, …, T_{d}) and a state ρ of C^{∗}(S) satisfying the conditions of Lemma 7.16. The set of operators {T_{1}, …, T_{d}} must be linearly independent; indeed, for every polynomial f ≠ 0 we have

ρ(f(T_{1}, …, T_{d})^{∗}f(T_{1}, …, T_{d})) = ∥f∥_{H^{2}}^{2} ≠ 0

and hence f(T_{1}, …, T_{d}) ≠ 0. It follows that span{T_{1}, …, T_{d}} = S.

The GNS construction provides a nondegenerate representation σ of C^{∗}(S) on a Hilbert space K and a unit vector ξ ∈ K such that ρ(X) = ⟨σ(X)ξ, ξ⟩, X ∈ C^{∗}(S). The key property of ρ implies that we can define an isometry U : H_{d}^{2} → K on polynomials by

Uf = σ(f(T_{1}, …, T_{d}))ξ, f ∈ P,

and we have US_{k} = σ(T_{k})U for every k = 1, …, d. Thus the range UH_{d}^{2} of U is invariant under each σ(T_{k}) and the restriction of the d-contraction (σ(T_{1}), …, σ(T_{d})) to UH_{d}^{2} is unitarily equivalent to the d-shift. By Lemma 7.14, the projection UU^{∗} must commute with σ(T_{k}), k = 1, …, d, and hence with the unital C^{∗}-algebra σ(C^{∗}(S)) these operators generate.

We obtain a representation π : C^{∗}(S) → B(H_{d}^{2}) by setting π(X) = U^{∗}σ(X)U. Since π(T_{k}) = U^{∗}σ(T_{k})U = S_{k} for each k it follows that π(S) = S_{d}. □

8. Various applications.

In this section we give several applications of the preceding results to function theory and multivariable operator theory. These are a version of von Neumann’s inequality for arbitrary d-contractions, a model theory for d-contractions based on the d-shift, and a discussion of the absence of inner functions in the multiplier algebra of the d-shift.

We point out that Popescu has established various versions of von Neumann’s inequality for non-commutative d-tuples of operators [30,32,34,35]. Here, on the other hand, we are concerned with d-contractions. The version of von Neumann’s inequality that is appropriate for d-contractions is the following.

Theorem 8.1. Let T̄ = (T_{1}, …, T_{d}) be an arbitrary d-contraction acting on a Hilbert space H. Then for every polynomial f in d complex variables we have

∥f(T_{1}, …, T_{d})∥ ≤ ∥f∥_{M},

∥f∥_{M} being the norm of f in the multiplier algebra M of H^{2}.

More generally, let (S_{1}, …, S_{d}) be the d-shift and let A ⊆ T_{d} be the algebra of all polynomials in S_{1}, …, S_{d}. Then the representation f ↦ f(T_{1}, …, T_{d}) defines a completely contractive representation of A.

proof. The assertions are immediate consequences of Theorem 6.2, once one observes that ∥f∥_{M} = ∥f(S_{1}, …, S_{d})∥. □

Turning now to models, we first recall some of the literature of dilation theory in d-dimensions. There are a number of positive results concerning non-commutative models for non-commuting d-tuples which satisfy the conditions of remark 3.2. The first results along these lines are due to Frazho [21] for pairs of operators. Frazho’s results were generalized by Bunce [15] to d-tuples. Popescu has clarified that work by showing that such a d-tuple can often be obtained by compressing a certain natural d-tuple of isometries acting on the full Fock space F(ℂ^{d}) over ℂ^{d} (the left creation operators) to a co-invariant subspace of F(ℂ^{d}), and he has worked out a functional calculus for that situation [28,29,30,31]. We also point out some recent work of Davidson and Pitts [18,19], relating to the operator algebra generated by the left creation operators on the full Fock space.

There is relatively little in the literature of operator theory, however, that relates to uniqueness of dilations in higher dimensions (however, see [11]). Indeed, normal dilations for d-contractions, when they exist, are almost never unique. On the other hand, recent results in the theory of semigroups of completely positive maps do include uniqueness. Generalizing work of Parathasarathy, B. V. R. Bhat [14] has shown that a unital semigroup of completely positive maps of a von Neumann algebra M can be dilated uniquely to an E_{0}-semigroup acting on a larger von Neumann algebra N which contains M as a hereditary subalgebra. A similar (and simpler) result holds for single unital completely positive maps: there is a unique dilation to a unital endomorphism acting on a larger von Neumann algebra as above. In the case where M = B(H), the latter dilation theorem is closely related to the Bunce-Frazho theory of d-tuples by way of the metric operator space associated with a normal completely postive map of B(H) [8,9]. SeLegue [42] has succeeded in unifying these results.

In the following discussion, we reformulate Theorem 6.2 as a concrete assertion about d-contractions which parallels some of the principal assertions of the Sz.-Nagy Foias model theory of 1-contractions [43]. Much of Theorem 8.5 follows directly from Theorem 6.2 and standard lore on the representation theory of C^{∗}-algebras. For completeness, we have given a full sketch of the argument.

We recall some elementary facts about the representation theory of C^{∗}-algebras such as T_{d}. Let π : T_{d} → B(H) be a nondegenerate ∗-representation of T_{d} on a separable Hilbert space H. Because of the exact sequence of Theorem 5.7, standard results about the representations of the C^{∗}-algebra of compact operators imply that π decomposes into a direct sum π_{1} ⊕ π_{2} where π_{1} is a multiple of n = 0, 1, 2, …, ∞ copies of the identity representation of T_{d} and π_{2} is a representation which annihilates K. π_{1} and π_{2} are disjoint as representations of T_{d}. This decomposition is unique in the sense that if π′_{1} is another multiple of n′ copies of the identity representation of T_{d} and π′_{2} annihilates K, and if π′_{1} ⊕ π′_{2} is unitarily equivalent to π_{1} ⊕ π_{2}, then n′ = n and π′_{2} is unitarily equivalent to π_{2} [5].

We will make use of these observations in a form that relates more directly to operator theory.

Definition 8.2. Let d = 1, 2, . . .. By a spherical operator (of dimension d) we mean a d-tuple (Z_{1}, …, Z_{d}) of commuting normal operators acting on a common Hilbert space such that

Z_{1}^{∗}Z_{1} + ⋯ + Z_{d}^{∗}Z_{d} = 1.

Spherical operators are the higher dimensional counterparts of unitary operators. For every spherical operator (Z_{1}, …, Z_{d}) acting on H there is a unique unital ∗-representation π : C(∂B_{d}) → B(H) which carries the d-tuple of canonical coordinate functions to (Z_{1}, …, Z_{d}). This relation between d-dimensional spherical operators and nondegenerate representations of C(∂B_{d}) is bijective.

If T̄ = (T_{1}, …, T_{d}) is an arbitrary d-tuple of operators acting on a common Hilbert space H and n is a nonnegative integer or +∞ we will write n · T̄ = (n · T_{1}, …, n · T_{d}) for the d-tuple of operators acting on the direct sum of n copies of H defined by

n · T_{k} = \underbrace{T_{k} ⊕ T_{k} ⊕ …}_{n times},

where for n = 0 the left side is interpreted the nil operator, that is, no operator at all. The direct sum of two d-tuples of operators is defined in the obvious way as a d-tuple acting on the direct sum of Hilbert spaces. The preceding remarks are summarized as follows.

Proposition 8.3. Let (n, Z̄) be a pair consisting of an integer n = 0, 1, 2, …, ∞ and a spherical operator Z̄ = (Z_{1}, …, Z_{d}) (which may be the nil d-tuple when n ≥ 1). Then there is a unique nondegenerate representation π of T_{d} satisfying

π(S_{k}) = n · S_{k} ⊕ Z_{k}, k = 1, …, d.

Every nondegenerate representation of T_{d} on a separable Hilbert space arises in this way, and if (n′, Z̄′) is another such pair giving rise to a representation π′, then π′ is unitarily equivalent to π iff n′ = n and Z̄′ is unitarily equivalent to Z̄.

Remarks. Of course, if Z̄ is the nil d-tuple then its corresponding summand in the definition of π is absent. Let S ⊆ B(H) be a set of operators acting on a Hilbert space H. A subspace K ⊆ H is said to be co-invariant under S is S^{∗}K ⊆ K. K is co-invariant iff its orthogonal complement is invariant, SK^{⊥} ⊆ K^{⊥}. A co-invariant subspace K is called full if H is spanned by {T ξ: ξ ∈ K} where T ranges over the C^{∗}-algebra generated by S. The following are equivalent for any co-invariant subace K: (8.4.1) K is full. (8.4.2) H is the smallest reducing subspace for S which contains K. (8.4.3) For every operator T in the commutant of S ∪ S^{∗} we have

T K = {0} ⟹ T = 0.

Let A be the algebra generated by S and the identity. We will often have a situation in which the C^{∗}-algebra generated by A is spanned by the set of products AA^{∗}, and in that case the following criterion can be added to the preceding list (8.4.4) H is the smallest invariant subspace for S which contains K. Indeed, since C^{∗}(A) is spanned by AA^{∗} we have

\overline{span} C^{∗}(A)K = \overline{span} AA^{∗}K = \overline{span} AK,

hence (8.4.1) and (8.4.4) are equivalent.

Since the d-shift is a d-contraction, any d-tuple (T_{1}, …, T_{d}) of the form

T_{k} = n · S_{k} ⊕ Z_{k}

described in Proposition 8.3 is a d-contraction. If K is any co-invariant subspace for {T_{1}, …, T_{d}} then the d-tuple (T′_{1}, …, T′_{d}) obtained by compressing to K

T′_{k} = P_{K}T_{k} ↾_{K}

is also a d-contraction. Indeed, for each k = 1, …, d we have

T′_{k}T_{k}^{′∗} = P_{K}T_{k}P_{K}T_{k}^{∗} ↾_{K}≤ P_{K}T_{k}T_{k}^{∗} ↾_{K},

and therefore ∑_{k} T′_{k}T_{k}^{′∗} ≤ 1. The following implies that d-tuples obtained from this construction are the most general d-contractions.

Theorem 8.5. Let d = 1, 2, …, let T̄ = (T_{1}, …, T_{d}) be a d-contraction acting on a separable Hilbert space and let S̄ = (S_{1}, …, S_{d}) be the d-shift. Then there is a triple (n, Z̄, K) consisting of an integer n = 0, 1, 2, …, ∞, a spherical operator Z̄, and a full co-invariant subspace K for the operator

n · S̄ ⊕ Z̄

such that T̄ is unitarily equivalent to the compression of n · S̄ ⊕ Z̄ to K.

Let T′ = (T′_{1}, …, T′_{d}) be another d-contraction associated with another such triple (n′, Z̄′, K′). If T̄ and T̄′ are unitarily equivalent then n′ = n, and there are unitary operators V ∈ B(n · H^{2}) and W : HZ̄ → HZ̄′ such that for k = 1, …, d we have

V S_{k} = S_{k}V, W Z_{k} = Z′_{k}W,

and which relate K to K′ by way of (V ⊕ W)K = K′.

Finally, the integer n is the rank of the defect operator

1 − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗},

and Z̄ is the nil spherical operator iff T̄ is a null d-contraction.

Remark 8.6. Notice that the situation of (8.4.4) prevails in this case, and we may conclude that for the triple (n, Z̄, K) associated with T̄ by Theorem 8.5, the Hilbert space H̃ on which n · S̄ ⊕ Z̄ acts is generated as follows

H̃ = \overline{span}\{f(n · S_{1} ⊕ Z_{1}, …, n · S_{d} ⊕ Z_{d})ξ: ξ ∈ K, f ∈ P\}

P denoting the set of all polynomials in d complex variables.

Befire giving the proof of Theorem 8.5 we want to emphasize the following general observation which asserts that, under certain conditions, a unitary operator which intertwines two representations of a subalgebra A of a C^{∗}-algebra B can be extended to a unitary operator which intertwines ∗-representations of B.

We recall a general theorem of Stinespring, which asserts that every completely positive map

ϕ : B → B(H)

defined on a unital C^{∗}-algebra B can be represented in the form ϕ(x) = V^{∗}π(x)V, where π is a representation of B on another Hilbert space H_{π}, and V ∈ B(H, H_{π}). The pair (V, π) is called minimal if

H_{π} = \overline{span}[π(x)ξ : x ∈ B, ξ ∈ H].

One can always arrange that (V, π) is minimal by cutting down to a suitable subrepresentation of π.

Lemma 8.6. Let B be a C^{∗}-algebra and let A be a (perhaps non self-adjoint) subalgebra of B such that

(8.7)  B = \overline{span}^{∥·∥}AA^{∗}.

For k = 1, 2 let ϕ_{k} : B → B(H_{k}) be A-morphisms, and let U : H_{1} → H_{2} be a unitary operator such that

Uϕ_{1}(a) = ϕ_{2}(a)U, a ∈ A.

Let (V_{k}, π_{k}) be a minimal Stinespring pair for ϕ_{k}, ϕ_{k}(x) = V_{k}^{∗}π_{k}(x)V_{k}, x ∈ B. Then there is a unique unitary operator W : H_{π_{1}} → H_{π_{2}} such that

(i) W π_{1}(x) = π_{2}(x)W, x ∈ B and (ii) W V_{1} = V_{2}U.

proof. Since both ϕ_{1} and ϕ_{2} are A-morphisms, the hypothesis on U implies that Uϕ_{1}(ab^{∗}) = ϕ_{2}(ab^{∗})U for all a, b ∈ A. Hence (8.7) implies that Uϕ_{1}(x) = ϕ_{2}(x)U for every x ∈ B. The rest now follows from standard uniqueness assertions about minimal completely positive dilations of completely positive maps of C^{∗}-algebras [3] □

Remark. There are many examples of subalgebras A of C^{∗}-algebras B that satisfy (8.7) besides the algebra A of polynomials in the Toeplitz algebra T_{d}. Indeed, if A is any algebra of operators on a Hilbert space which satisfies

A^{∗}A ⊆ A + A^{∗}

then the linear span of AA^{∗} is closed under multiplication, and hence the normclosed linear span of AA^{∗} is a C^{∗}-algebra. Such examples arise in the theory of E_{0}-semigroups [6], and in the Cuntz C^{∗}-algebras O_{n}, n = 2, …, ∞.

proof of Theorem 8.5. Suppose that the operators T_{k} act on a Hilbert space H. Let A be the algebra of all polynomials in the d-shift S̄ = (S_{1}, …, S_{d}). By Theorem 6.3 there is an A-morphism

ϕ : T_{d} → B(H)

such that ϕ(S_{k}) = T_{k} for k = 1, …, d. Let

ϕ(X) = V^{∗}π(X)V, X ∈ T_{d}

be a minimal Stinespring representation of ϕ. We have

V^{∗}V = V^{∗}π(1)V = ϕ(1) = 1,

hence V is an isometry.

We claim that V H is co-invariant under π(A),

(8.8)  π(A)^{∗}V H ⊆ V H.

Indeed, if A ∈ A and P denotes the projection P = V V^{∗} then for every X ∈ T_{d} we have

P π(A)P π(X)V = V ϕ(A)ϕ(X) = V ϕ(AX) = P π(AX)V = P π(A)π(X)V

and hence the operator P π(A)P − P π(A) vanishes on

\overline{span}[π(X)ξ : X ∈ T_{d}, ξ ∈ H] = H_{π}.

Thus π(A)^{∗}P = P π(A)^{∗}P and (8.8) follows.

Because of minimality of (V, π) it follows that the subspace K = V H ⊆ H_{π} is a full co-invariant subspace for the operator algebra π(A).

Proposition 8.3 shows that if we replace π with a unitarily equivalent representation and adjust V accordingly then we may assume that there is an integer n = 0, 1, 2, …, ∞ and a (perhaps nil) spherical operator Z̄ = (Z_{1}, …, Z_{d}) such that H_{π} = n · H^{2} ⊕ HZ̄ and

π(S_{k}) = n · S_{k} ⊕ Z_{k}, k = 1, …, d.

That proves the first paragraph of Theorem 8.5.

The second paragraph follows after a straightforward application of Lemma 8.6, once one notes that if we are given two triples (n, Z̄, K) and (n′, Z̄′, K′) and we define representations π and π′ of T_{d} by

π(S_{k}) = n · S_{k} ⊕ Z_{k} = σ_{1}(S_{k}) ⊕ σ_{2}(S_{k}),
π′(S_{k}) = n′ · S_{k} ⊕ Z′_{k} = σ′_{1}(S_{k}) ⊕ σ′_{2}(S_{k}),

then σ_{1} is disjoint from σ_{2}, σ′_{1} is disjoint from σ′_{2}, while σ_{k} is quasi-equivalent to σ′_{k}. Thus, any unitary operator W which intertwines the representations π and π′ must decompose into a direct sum W = W_{1} ⊕ W_{2} where W_{1} intertwines σ_{1} and σ′_{1} and W_{2} intertwines σ_{2} and σ′_{2}.

To prove the third paragraph, choose an integer n = 0, 1, 2, …, ∞, let Z̄ = (Z_{1}, …, Z_{d}) be a spherical operator whose component operators act on a Hilbert space L, and let K ⊆ n · H^{2} ⊕ L be a full co-invariant subspace for the operator

n · S̄ ⊕ Z̄,

where S̄ = (S_{1}, …, S_{d}) is the d-shift. Define T̄ = (T_{1}, …, T_{d}) by

T_{j} = P_{K}(n · S_{j} ⊕ Z_{j}) ↾_{K},

j = 1, …, d. We have to identify the multiplicity n and the existence of the spherical summand Z̄ in terms of T̄.

Let P_{K} ∈ B(n · H^{2} ⊕ L) denote the projection on K. Since K is co-invariant under n · S̄ ⊕ Z̄ we have

P_{K}(n · S_{j} ⊕ Z_{j}) = P_{K}(n · S_{j} ⊕ Z_{j})P_{K} = T_{j}P_{K}

for every j = 1, …, d, and hence

(8.9)  T_{j}T_{j}^{∗} = P_{K}(n · S_{j}S_{j}^{∗} ⊕ Z_{j}Z_{j}^{∗}) ↾_{K} .

By the remarks following Definition 2.10 we may sum on j to obtain

(8.10)  ∑_{j=1}^{d} T_{j}T_{j}^{∗} = P_{K}(n · (1 − E_{0}) ⊕ 1_{L}) ↾_{K}= 1_{K} − P_{K}(n · E_{0} ⊕ 0) ↾_{K}

where E_{0} ∈ B(H^{2}) denotes the one-dimensional projection onto the constants. From (8.10) we find that the defect operator D has the form

(8.11)  D = 1_{K} − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗} = P_{K}(n · E_{0} ⊕ 0) ↾_{K} .

Now for any positive operator B we have Bξ = 0 iff ⟨Bξ, ξ⟩ = 0. Thus the relation (8.11) between the positive operators D and n · E_{0} ⊕ 0 implies that their kernels are related by

{ξ ∈ K : Dξ = 0} = {ξ ∈ K : (n · E_{0} ⊕ 0)ξ = 0},

and hence

rankD = dim((n · E_{0} ⊕ 0)K).

The dimension of the space N = (n · E_{0} ⊕ 0)K is easily seen to be n. Indeed, notice that if A ∈ B(H^{2}) is a polynomial in the operators S_{1}, …, S_{d} then we have E_{0}A = E_{0}AE_{0} = ⟨A1, 1⟩ E_{0}, and hence E_{0}A is a scalar multiple of E_{0}. Similarly, if B ∈ B(n · H^{2} ⊕ L) is a polynomial in the operators n · S_{1} ⊕ Z_{1}, …, n · S_{d} ⊕ Z_{d} then (n · E_{0} ⊕ 0)B is a scalar multiple of (n · E_{0} ⊕ 0), and hence for all such B we have

(n · E_{0} ⊕ 0)BK ⊆ N.

Because K is a full co-invariant subspace, (8.4.4) implies that n · H^{2} ⊕ L is spanned by vectors of the form Bξ, with B as above and ξ ∈ K. It follows that

(n · E_{0} ⊕ 0)(n · H^{2} ⊕ L) ⊆ N,

and therefore N is the range of the n-dimensional projection n · E_{0} ⊕ 0. Hence dim N = n.

Finally, we consider case in which T̄ is a null d-contraction. Let Q and P be the completely positive maps on B(H^{2}) and B(K) given respectively by

P (A) = S_{1}AS_{1}^{∗} + ⋯ + S_{d}AS_{d}^{∗},  A ∈ B(H^{2}),
Q(B) = T_{1}BT_{1}^{∗} + ⋯ + T_{d}BT_{d}^{∗},  B ∈ B(K).

Formula (8.9) implies that Q(1_{K}) = P_{K}(n · P (1_{H^{2}}) ⊕ 1_{L}) ↾_{K}. Similarly, using co-invariance of K repeatedly as in (8.9) we have

T_{j_{1}} … T_{j_{r}}T_{j_{r}}^{∗} … T_{j_{1}}^{∗} = P_{K}(n · (S_{j_{1}} … S_{j_{r}}S_{j_{r}}^{∗} ⋯ S_{j_{1}}^{∗}) ⊕ Z_{j_{1}} ⋯ Z_{j_{r}}Z_{j_{r}}^{∗} ⋯ Z_{j_{1}}^{∗}) ↾_{K}

for every j_{1}, …, j_{r} ∈ {1, …, d}. After summing on j_{1}, …, j_{r} we obtain

Q^{r}(1_{K}) = P_{K}(n · P^{r}(1_{H^{2}}) ⊕ 1_{L}) ↾_{K}, r = 1, 2, ….

Since P^{r}(1_{H^{2}}) ↓ 0 as r → ∞, we have

lim_{r→∞} Q^{r}(1_{K}) = P_{K}(0 ⊕ 1_{L}) ↾_{K} .

We conclude that T̄ is a null d-tuple iff 0 ⊕ L ⊥ K, that is, K ⊆ n · H^{2} ⊕ {0}. Noting that n · H^{2} ⊕ {0} is a reducing subspace for the operator n · S̄ ⊕ Z̄ we see from (8.4.2) that

n · H^{2} ⊕ L ⊆ n · H^{2} ⊕ {0},

and therefore L = {0}. But a spherical d-tuple cannot be the zero d-tuple except when it is the nil d-tuple, and thus we have proved that T̄ is a null d-contraction iff Z̄ is nil □

The two extreme cases of Theorem 8.5 in which n = 0 and n = 1 are noteworthy. From the case n = 0 we deduce the following result of Athavale [11], which was established by entirely different methods.

Corollary 1. Let T_{1}, …, T_{d} be a set of commuting operators on a Hilbert space H such that T_{1}^{∗}T_{1} + ⋯ + T_{d}^{∗}T_{d} = 1. Then (T_{1}, …, T_{d}) is a subnormal d-tuple.

proof. Let A_{k} = T_{k}^{∗}. (A_{1}, …, A_{d}) is a d-contraction for which

n = rank(1 − A_{1}A_{1}^{∗} − ⋯ − A_{d}A_{d}^{∗}) = 0.

Theorem 8.5 implies that there is a spherical operator Z̄ = (Z_{1}, …, Z_{d}) acting on a Hilbert space H̃ ⊇ H such that Z_{k}^{∗}H ⊂ H and A_{k} is the compression of Z_{k} to H, k = 1, …, d. Hence T_{k} = A_{k}^{∗} = Z_{k}^{∗} ↾_{H} for every k, so that (Z_{1}^{∗}, …, Z_{d}^{∗}) is a normal d-tuple which extends (T_{1}, …, T_{d}) to a larger Hilbert space □

From the case n = 1 we have the following description of all d-contractions that can be obtained by compressing the d-shift to a co-invariant subspace.

Corollary 2. Every nonzero co-invariant subspace K ⊆ H^{2} for the d-shift S̄ = (S_{1}, …, S_{d}) is full, and the compression of S̄ to K

T_{k} = P_{K}S_{k} ↾_{K}, k = 1, …, d

defines a null d-contraction T̄ = (T_{1}, …, T_{d}) for which

(8.12)  rank(1 − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗}) = 1.

If K′ is another co-invariant subspace for S̄ which gives rise to T̄′, then T̄ and T̄′ are unitarily equivalent if and only if K = K′.

Every null d-contraction (T_{1}, …, T_{d}) satisfying (8.12) is unitarily equivalent to one obtained by compressing (S_{1}, …, S_{d}) to a co-invariant subspace of H^{2}.

proof. Let {0} ≠ K ⊆ H^{2} be a a co-invariant subspace for the set of operators {S_{1}, …, S_{d}}. Since T_{d} is an irreducible C^{∗}-algebra it follows that K satisfies condition (8.4.2), hence it is full. Let T_{j} be the compression of S_{j} to K, j = 1, …, d. The canonical triple associated with T̄ = (T_{1}, …, T_{d}) is therefore (1, nil, K), and the third paragraph of Theorem 8.5 implies that T̄ is a null d-contraction satisfying (8.12).

If K′ is another co-invariant subspace of H^{2} giving rise to a d-contraction T̄′ which is unitarily equivalent to T̄ then Theorem 8.5 implies that there is a unitary operator V which commutes with S = {S_{1}, …, S_{d}} such that V K = K′. Because V is unitary it must commute with S^{∗} as well, and hence with the Toeplitz algebra T_{d}. The latter is irreducible, hence V must be a scalar multiple of the identity operator, hence K′ = K.

Finally, if T̄ = (T_{1}, …, T_{d}) is any null d-contraction then Theorem 8.5 implies that the spherical summand Z̄ of its dilation must be the nil d-tuple, and if in addition

rank(1 − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗}) = 1,

then the canonical triple associated with T̄ is (1, nil, K) for some subspace K of H^{2} which is co-invariant under the d-shift □

Lemma 7.13 asserts that the identity representation of the Toeplitz C^{∗}-algebra is a boundary representation for the unital operator space generated by the d-shift. This fact has a number of interesting consequences, and we conclude with a brief discussion of one of them. Rudin posed the following function-theoretic problem in the sixties: Do there exist nonconstant inner functions in H^{∞}(B_{d}) [37]? This problem was finally solved (affirmatively) in 1982 by B. A. Aleksandrov [38]. The following Proposition implies that the answer to the analogue of Rudin’s question for the multiplier algebra M is the opposite: there are no nontrivial isometries in B(H^{2}) which commute with {S_{1}, …, S_{d}} when d ≥ 2. Indeed, we have the following more general assertion.

Proposition 8.13. Let T_{1}, T_{2}, … be a finite or infinite sequence of operators on H_{d}^{2}, d ≥ 2, which commute with the d-shift and which satisfy

(8.6)  T_{1}^{∗}T_{1} + T_{2}^{∗}T_{2} + ⋯ = 1.

Then each T_{j} is a scalar multiple of the identity operator.

proof. Consider the completely positive linear map ϕ defined on B(H^{2}) by

ϕ(A) = T_{1}^{∗}AT_{1} + T_{2}^{∗}AT_{2} + ….

The sum converges strongly for every operator A because by (8.6) we have

∥T_{1}ξ∥^{2} + ∥T_{2}ξ∥^{2} + ⋯ = ∥ξ∥^{2} < ∞, ξ ∈ H^{2}.

Moreover, since each T_{k} commutes with each S_{j} we have T_{k}^{∗}S_{j}T_{k} = T_{k}^{∗}T_{k}S_{j}, and thus from (8.6) we conclude that ϕ(A) = A for every A in S = span{1, S_{1}, …, S_{d}}. Since the identity representation of T_{d} is a boundary representation for S it follows that ϕ(A) = A for every A in the Toeplitz C^{∗}-algebra T_{d}.

Let n be the number of operators in the sequence T_{1}, T_{2}, … and let V be the linear map of H^{2} to n · H^{2} defined by

V ξ = (T_{1}ξ, T_{2}ξ, …).

Because of (8.6) V is an isometry. Letting π be the representation of B(H^{2}) on n · H^{2} defined by

π(A) = A ⊕ A ⊕ …,

we find that (V, π) is a Stinespring pair for ϕ,

ϕ(A) = V^{∗}π(A)V, A ∈ B(H^{2}).

Since

(V A − π(A)V)^{∗}(V A − π(A)V) = A^{∗}ϕ(1)A − ϕ(A)^{∗}A − A^{∗}ϕ(A) + ϕ(A^{∗}A) = 0,

we conclude that V A − π(A)V = 0. By examining the components of this operator equation one sees that T_{k}A = AT_{k} for every k and every A ∈ T_{d}. Since T_{d} is an irreducible C^{∗}-algebra it follows that each T_{k} must be a scalar multiple of the identity operator □

Appendix. Trace estimates.

Fix d = 1, 2, …, let E_{d} be a d-dimensional Hilbert space, and let

F_{+}(E_{d}) = ℂ ⊕ E_{d} ⊕ E_{d}^{2} ⊕ …

be the symmetric Fock space over E_{d}. The number operator is the unbounded self-adjoint diagonal operator N satisfying Nξ = nξ, ξ ∈ E_{d}^{n}, n = 0, 1, . . .. Let P_{n} be the projection on E_{d}^{n}. Then for every p > 0, (1 + N)^{−p} is a positive compact operator

(1 + N)^{−p} = ∑_{n=0}^{∞}(n + 1)^{−p}P_{n}

whose trace is given by

(A.1)  trace(1 + N)^{−p} = ∑_{n=0}^{∞} \frac{dim E_{d}^{n}}{(n + 1)^{p}} .

Thus (1 + N)^{−1} belongs to the Schatten class L^{p}(F_{+}(E_{d})) iff the infinite series (A.1) converges. In this appendix we show that is the case iff p > d. Notice that the function of a complex variable defined for ℜz > d by

ζ_{d}(z) = trace(1 + N)^{−z}

is a d-dimensional variant of the Riemann zeta function, since for d = 1 we have dim E_{d}^{n} = 1 for all n and hence

ζ_{1}(z) = ∑_{n=1}^{∞} \frac{1}{n^{z}} .

We calculate the generating function for the coefficients dim E_{d}^{n}.

Lemma A.2. The numbers a_{n,d} = dim E_{d}^{n} are the coefficients of the series expansion

(1 − z)^{−d} = ∑_{n=0}^{∞} a_{n,d}z^{n}, |z| < 1.

proof. Note that the numbers a_{n,d} satisfy the recurrence relation

(A.3)  a_{n,d+1} = a_{0,d} + a_{1,d} + ⋯ + a_{n,d},  n = 0, 1, …, d = 1, 2, ….

Indeed, if we choose a basis e_{1}, …, e_{d} for E_{d} then the set of symmetric products

{e_{i_{1}}e_{i_{2}} … e_{i_{n}} : 1 ≤ i_{1} ≤ ⋯ ≤ i_{n} ≤ d}

forms a basis for the vector space E_{d}^{n} and hence a_{n,d} is the cardinality of the set

S_{n,d} = {(i_{1}, …, i_{n}) ∈ {1, …, d}^{n} : 1 ≤ i_{1} ≤ ⋯ ≤ i_{n} ≤ d}.

Since S_{n,d+1} decomposes into a disjoint union

S_{n,d+1} = ⨆_{k=0}^{n}{(i_{1}, …, i_{n}) ∈ S_{n,d+1} : i_{k} ≤ d, i_{k+1} = ⋯ = i_{n} = d + 1},

and since the kth set on the right has the same cardinality a_{k,d} as S_{k,d}, (A.3) follows.

From (A.3) we find that a_{n,d+1} −a_{n−1,d+1} = a_{n,d}. Thus if we let f_{d} be the formal power series

(A.4)  f_{d}(z) = ∑_{n=0}^{∞} a_{n,d}z^{n}

then f_{d+1}(z) − zf_{d+1}(z) = f_{d}(z), hence

f_{d+1}(z) = \frac{f_{d}(z)}{1 − z}.

Lemma A.1 follows after noting that f_{1}(z) = 1 + z + z^{2} + ⋯ = (1 − z)^{−1} □

Remark. Notice that the power series of Lemma A.2 converges absolutely to the generating function (1 − z)^{−d} throughout the open unit disk |z| < 1.

By evaluating successive derivatives of the generating function at the origin, we find that

(A.5)  dim E_{d}^{n} = \frac{(n + d − 1)(n + d − 2) … d}{n!} = \frac{(n + d − 1)!}{n!(d − 1)!}.

A straightforward application of Stirling’s formula [36, p. 194]

N! ∼ \sqrt{2π} N^{N+1/2}e^{−N}

leads to

lim_{n→∞}(n + 1)^{−d+1} \frac{(n + d − 1)!}{n!} = \frac{1}{(d − 1)!}

and hence

(A.6)  dim E_{d}^{n} ∼ \frac{(n + 1)^{d−1}}{(d − 1)!} .

We now prove the assertion of (5.2).

Theorem. For p > 0 we have trace(1 + N)^{−p} < ∞ if and only if p > d.

proof. By (A.6), the infinite series

trace(1 + N)^{−p} = ∑_{n=0}^{∞} \frac{dim E_{d}^{n}}{(n + 1)^{p}}

converges if and only if the series

∑_{n=0}^{∞} \frac{1}{(n + 1)^{p−d+1}}

converges; i.e., iff p > d □

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