C^{∗}-ALGEBRAS AND NUMERICAL LINEAR ALGEBRA

William Arveson

Department of Mathematics University of California Berkeley, CA 94720 USA

Abstract. Given a self adjoint operator A on a Hilbert space, suppose that that one wishes to compute the spectrum of A numerically. In practice, these problems often arise in such a way that the matrix of A relative to a natural basis is “sparse”. For example, discretized second order differential operators can be represented by doubly infinite tridiagonal matrices. In these cases it is easy and natural to compute the eigenvalues of large n × n submatrices of the infinite operator matrix, and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of A. Numerical analysts call this the Galerkin method.

While this hope is often realized in practice it often fails as well, and it can fail in spectacular ways. The sequence of eigenvalue distributions may not converge as n → ∞, or they may converge to something that has little to do with the original operator A. At another level, even the meaning of ‘convergence’ has not been made precise in general. In this paper we determine the proper general setting in which one can expect convergence, and we describe the asymptotic behavior of the n × n eigenvalue distrubutions in all but the most pathological cases. Under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of A is recovered from the sequence of eigenvalues of the n × n compressions.

In broader terms, our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms of C^{∗}- algebras. Indeed, it is only when the single operator A is viewed as an element of an appropriate C^{∗}-algebra A that one can see the precise nature of the limit of the n×n eigenvalue distributions; the limit is associated with a tracial state on A. Normally, A is highly noncommutative, and in our main applications it is a simple C^{∗}-algebra having a unique tracial state. We obtain precise asymptotic results for operators which represent discretized Hamiltonians of one-dimensional quantum systems with arbitrary continuous potentials.

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. spectrum, eigenvalues, Szegö’s theorem. This research was supported in part by NSF grant DMS89-12362

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1. Introduction.

There are efficient algorithms available for computing the spectrum of self-adjoint n × n matrices. Using a good desktop computer, one can find all of the eigenvalues of a 100×100 Toeplitz matrix in a few seconds. It is natural to ask how these finitedimensional methods might be applied to compute the spectrum of the discretized Hamiltonian of a one dimensional quantum system, or a self-adjoint operator in an irrational rotation C^{∗}-algebra. Considering the burgeoning capabilities of modern computers, it is unfortunate that there are apparently no guidelines available which indicate how one might proceed with such a project except in isolated cases (I am indebted to the referee for providing a good example [13]). For instance, we were unable to determine from the literature if such a program is even possible in general.

The purpose of this paper is to develop some general methods for computing the spectrum (more precisely, the essential spectrum) of a self-adjoint operator in terms of the eigenvalues of a sequence of finite dimensional matrix approximations. We will see that there is a natural notion of degree of an operator (relative to a filtration of the underlying Hilbert space into finite dimensional subspaces) which generalizes the classical idea of band-limited matrices. More significantly, we will find that this kind of numerical analysis leads one to a broader context, in which the operator is seen as an element of a C^{∗}-algebra which is frequently simple and possesses at least one tracial state. The limit distributions arising from the finite dimensional eigenvalue distributions correspond to traces on this C^{∗}-algebra.

Our intention is not to discuss algorithms per se, but rather to present an effective program for utilizing the finite dimensional techniques to compute infinite dimensional spectra, along with a precise description of a broad class of operators for which it will succeed. Thus, we deal with issues arising from operator theory and operator algebras; we do not address questions relating directly to numerical analysis.

In more concrete terms, given an orthonormal basis {e_{1}, e_{2}, … } for a Hilbert space H and an operator A ∈ B(H), we may consider the associated infinite matrix (a_{ij})

a_{ij} =< Ae_{j}, e_{i} >, i, j = 1, 2, …,

and one can attempt to approximate the spectrum of A by calculating the eigenvalues of large n × n submatrices

A_{n} = ( \begin{matrix}a_{11} & a_{12} & … & a_{1n} \\ a_{21} & a_{22} & … & a_{2n} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{n1} & a_{n2} & … & a_{nn} \end{matrix} ) .

This program fails dramatically in general. For example, if A is the simple unilateral shift (defined on such a basis by Ae_{n} = e_{n+1}, n ≥ 1), then the spectrum of A is the closed unit disk and the essential spectrum of A is the unit circle, while σ(A_{n}) = {0} for every n because each A_{n} is a nilpotent matrix.

Examples like this have caused operator theorists to view matrix approximations with suspicion, and to routinely look elsewhere for effective means of computing spectra. On the other hand, since the 1920s there has been evidence that for selfadjoint operators, this program can be successfully carried out.

Consider for example the following theorem of Szegö. Let f be a real function in L^{∞}[−π, π] and let A be the multiplication operator

Aξ(x) = f(x)ξ(x), ξ ∈ L^{2}[−π, π].

With respect to the familiar orthonormal basis {e_{n} : n ∈ ℤ}, e_{n}(x) = e^{inx}, the matrix of A is a doubly-infinite Laurent matrix whose principal n × n submatrices are finite self-adjoint Toeplitz matrices

A_{n} = ( \begin{matrix}a_{0} & a_{−1} & … & a_{−n+1} \\ a_{1} & a_{0} & … & a_{−n+2} \\ ⋮ & & ... & ⋮ \\ a_{n−1} & … & & a_{0} \end{matrix} ) ,

the numbers a_{k} being the Fourier coefficients of f. Let {λ_{1}, λ_{2}, …, λ_{n}} be the eigenvalues of A_{n}, repeated according to multiplicity. We abuse notation slightly here in supressing the dependence of the individual eigenvalues on the parameter n, but this will not pose serious difficulties. Szegö’s theorem asserts that for every continuous function u : ℝ → ℝ we have

1.1  lim_{n→∞} \frac{1}{n}(u(λ_{1}) + u(λ_{2}) + ⋯ + u(λ_{n})) = \frac{1}{2π} ∫_{−π}^{π} u(f(x)) dx.

See [8][10]; a discussion of Szegö’s theorem can be found in [11], and [6] contains a meticulous discussion of Toeplitz operators.

Let us reformulate Szegö’s theorem in terms of weak^{∗}-convergence of measures. Let a and b be the essential inf and essential sup of f

a = ess inf_{−π≤x≤π} f(x)
b = ess sup_{−π≤x≤π} f(x),

and let μ be the probability measure defined on [a, b] by

μ(S) = \frac{1}{2π} m\{x : f(x) ∈ S\},

m denoting Lebesgue measure on [−π, π]. Then 1.1 asserts that the sequence of discrete probability measures

μ_{n} = \frac{1}{n}(δ_{λ_{1}} + δ_{λ_{2}} + ⋯ + δ_{λ_{n}}),

δ_{λ} denoting the unit point mass at λ, converges to μ in the weak^{∗}-topology of the dual of C[a, b],

lim_{n→∞} ∫_{a}^{b} u(x) dμ_{n}(x) = ∫_{a}^{b} u(x) dμ(x), u ∈ C[a, b].

The spectrum of A coincides with its essential spectrum in this case, and is exactly the closed support of the measure μ.

Once one knows that the measures μ_{n} converge in this way to a measure supported exactly on σ_{e}(A), one may draw rather precise conclusions about the rate at which the eigenvalues of A_{n} accumulate at points of the essential spectrum. For example, suppose that we choose a point λ ∈ σ_{e}(A). Then for every open interval I containing λ we have μ(I) > 0. If we choose positive numbers α and β which are close to μ(I) and satisfy α < μ(I) < β, then in the generic case where A has no point spectrum we may conclude that

nα < #(σ(A_{n}) ∩ I) < nβ

when n is sufficiently large. Of course, the symbol # means the number of eigenvalues in the indicated set, counting repeated eigenvalues according to the multiplicity of their occurrence. In a similar way, the density of eigenvalues of A_{n} tends to become increasingly sparse around points of the complement of the essential spectrum.

In this paper we will show that, while convergence of this type cannot be expected in general, it does persist in a much broader context than the above. Actually, this work was initiated in order to develop methods for computing the essential spectrum of certain tridiagonal operators which are associated with the discretized Hamiltonians of one-dimensional quantum mechanical systems (see section 5). These operators serve to illustrate the flavor of our more general results. Starting with a real valued continuous function V of a real variable, we fix a real number θ and consider the tridiagonal operator T defined in terms of a bilateral orthonormal basis {e_{n} : n ∈ ℤ} by

T e_{n} = e_{n−1} + d_{n}e_{n} + e_{n+1},

where the diagonal sequence is defined by d_{n} = V (sin(nθ)), n ∈ ℤ. The compression T_{n} of T to the span of {e_{k} : −n ≤ k ≤ n} is a (2n + 1) × (2n + 1) tridiagonal matrix whose eigenvalues can be computed very rapidly using modern algorithms. For example, we show that if μ_{n} is the discrete probability measure defined by the eigenvalue distribution of T_{n},

μ_{n} = \frac{1}{2n + 1}(δ_{λ_{1}} + δ_{λ_{2}} + … δ_{λ_{2n+1}}),

then there is a probability measure μ having closed support σ_{e}(T), such that for every continuous function f : ℝ → ℝ we have

lim_{n→∞} ∫ f(x) dμ_{n}(x) = ∫ f(x) dμ(x).

See formula 5.1.

In these cases the limiting measure μ can be identified in rather concrete terms. Briefly, if θ/π is irrational then the pair of unitary operators

Ue_{n} = e_{n+1}
V e_{n} = e^{inθ}e_{n}

generate an irrational rotation C^{∗}-algebra A_{θ}. A_{θ} has an essentially unique representation in which its weak closure is the II_{1} factor R. T becomes a self-adjoint operator in R, and the measure μ is defined by

μ(S) = τ(E_{T} (S)), S ⊆ ℝ

where τ is the normalized trace on R and E_{T} is the spectral measure of T.

In sections 2 and 3 we consider a rather general question. Given a self-adjoint operator A and an orthonormal basis for the underlying Hilbert space, let (a_{ij}) be the matrix of A relative to this basis. How must one choose the basis so that the essential spectrum of A can be computed in principle from the eigenvalues of the finite dimensional principal submatrices of (a_{ij})? Roughly speaking, we show that this is possible if the diagonals of the matrix (a_{ij}) tend to zero sufficiently fast. It is not possible in general. The family of all operators whose matrices satisfy this growth condition is a Banach ∗-algebra but it is not a C^{∗}-algebra.

In section 4 we seek more precise results which relate to weak^{∗}-convergence as above. It is appropriate to formulate these results in terms of C^{∗}-algebras of operators. We introduce the notion of degree of an operator (relative to a filtration of the underlying Hilbert space into finite dimensional subspaces), and show that one can expect convergence of this type in the state space of the given C^{∗}-algebra when the C^{∗}-algebra contains a dense set of finite degree operators and has a unique tracial state. These results are applied in section 5 to the discretized Hamiltonians described above.

2. Spectral asymptotics of sequences of matrices.

Let A be a bounded self-adjoint operator on a Hilbert space H and suppose that we have a sequence of finite-dimensional subspaces H_{n} of H with the property that the corresponding sequence of projections P_{n} ∼ H_{n} converges strongly to the identity. Normally, the subspaces H_{n} will be increasing with n, the main examples arising in the cases where we start with an orthonormal basis {e_{n} : n = 1, 2, … } (resp. {e_{n} : n ∈ ℤ}) and choose H_{n} = [e_{1}, e_{2}, …, e_{n}] (resp. H_{n} = [e_{−n}, e_{−n+1}, …, e_{n}]}).

For every n ≥ 1, let A_{n} denote the compression

2.1  A_{n} = P_{n}A|_{H_{n}}.

More generally, given an arbitrary sequence A_{1}, A_{2}, … of finite dimensional selfadjoint operators, we introduce two “spectral” invariants Λ, Λ_{e} of the sequence as follows. Λ is defined as the set of all λ ∈ ℝ with the property that there is a sequence λ_{1}, λ_{2}, … such that λ_{n} ∈ σ(A_{n}) and

lim_{n→∞} λ_{n} = λ.

Notice that Λ is closed. Indeed, a real number λ belongs to the complement of Λ iff there is an open set U containing λ and an infinite sequence of integers n_{1} < n_{2} < … with the property that σ(A_{n_{k}})∩U = ∅ for every k = 1, 2, . . .. Every point of U shares this property with λ, and hence the complement of Λ is open. If the sequence {A_{n}} is bounded in norm then Λ is compact. Of course, Λ is considerably smaller than the set

L = ∩_{n}(\overline{σ(A_{n}) ∪ σ(A_{n+1}) ∪ …})

of limit points of the sequence σ(A_{1}), σ(A_{2}), …, since limits along subsequences may not qualify for membership in Λ.

For every n ≥ 1 and every set S of real numbers, let N_{n}(S) be the number of eigenvalues of A_{n} which belong to S, counting multiple eigenvalues according to their multiplicity. N_{n}(·) is an integer-valued measure which takes values 0, 1, 2, …, dim H_{n}. The points of Λ are classified as follows.

Definition 2.2.

(1) A point λ ∈ ℝ is called essential if, for every open set U containing λ, we

have

lim_{n→∞} N_{n}(U) = ∞.

The set of essential points is denoted Λ_{e}.

(2) λ ∈ ℝ is called transient if there is an open set U containing λ such that

sup_{n≥1} N_{n}(U) < ∞.

Remarks. It is clear that Λ_{e} ⊆ Λ. Moreover, notice that λ ∈ ℝ is nonessential iff there is an open set U containing λ and an infinite sequence of integers n_{1} < n_{2} < … such that

N_{n_{k}}(U) ≤ M < ∞

for every k = 1, 2, . . .. Thus the nonessential points are an open set in ℝ, and we conclude that the set Λ_{e} of essential points is closed.

It is conceivable that Λ may contain anomalous points which are neither transient nor essential. In small neighborhoods U of such a point one would find subsequences n_{k} for which N_{n_{k}}(U) remains bounded, and other subsequences m_{k} for which N_{m_{k}}(U) is unbounded. Fortunately, in most situations one has a more manageable dichotomy (see Theorem 3.8). At this level of generality, one cannot say much. We do have the following

Theorem 2.3. Assuming that the sequence A_{1}, A_{2}, … arises from an operator A as in 1.1, then we have σ(A) ⊆ Λ and σ_{e}(A) ⊆ Λ_{e}.

proof of σ(A) ⊆ Λ.

Suppose that λ is a real number which does not belong to Λ. We will show that A−λ1 is invertible. Since λ ∉ Λ, there is an ϵ > 0 and a subsequence n_{1} < n_{2} < … such that

σ(A_{n_{k}}) ∩ (λ − ϵ, λ + ϵ) = ∅

for every k ≥ 1. Thus the distance from λ to σ(A_{n_{k}}) is at least ϵ, hence

∥ (A_{n_{k}} − λ1)^{−1} ∥≤ 1/ϵ

for every k. Let

B_{k} = (A_{n_{k}} − λ1)^{−1}P_{n_{k}}.

Since B_{1}, B_{2}, … is a bounded sequence of self-adjoint operators and the ball of B(H) of radius 1/ϵ is weakly sequentially compact, {B_{k}} has a weakly convergent subsequence. Thus, by replacing the original sequence n_{1}, n_{2}, … with a subsequence of itself, we may assume that there is a bounded operator B such that

lim_{k→∞} < B_{k}ξ, η >=< Bξ, η >, ξ, η ∈ H.

Since

2.4  (A_{n_{k}} − λ1)B_{k} = P_{n_{k}}

for every k and since (A_{n_{k}} − λ1)P_{n_{k}} converges to A − λ1 in the strong operator topology, it follows that the product (A_{n_{k}} −λ1)B_{n_{k}} converges weakly to (A−λ1)B. Hence from 2.4 we may conclude that

(A − λ1)B = weak lim_{k→∞} P_{n_{k}} = 1,

proving that A − λ1 is right-invertible. Left-invertiblity follows by taking adjoints in the preceding equation.

proof of σ_{e}(A) ⊆ Λ_{e}.

Suppose that λ ∉ Λ_{e}. We will show that A − λ1 is invertible modulo trace class operators, and hence λ ∉ σ_{e}(A). By the hypothesis on λ, there is a sequence n_{1} < n_{2} < … of positive integers and a pair of positive numbers ϵ, M such that

N_{n_{k}}(λ − ϵ, λ + ϵ) ≤ M

for every k = 1, 2, . . .. For each k, consider A_{n_{k}} to be a self-adjoint operator in B(P_{n_{k}}H) and let E_{k} be the spectral measure of this operator. Then Q_{k} = E_{k}(λ − ϵ, λ + ϵ) is a projection commuting with A_{n_{k}} satisfying dimQ_{k} ≤ M, such that the restriction B_{k} of A_{n_{k}} −λ1 to the range of P_{n_{k}} −Q_{k} is an invertible operator with ∥ B_{k}^{−1} ∥≤ 1/ϵ.

By using compactness again and passing to a subsequence as in the preceding argument, we can assume that both sequences {B_{k}^{−1}(P_{n_{k}} −Q_{k})} and {Q_{k}} converge in the weak operator topology

B_{k}^{−1}(P_{n_{k}} − Q_{k}) → C
Q_{k} → Q.

Notice that Q is a positive trace class operator. Indeed, since the set

{T ∈ B(H) : T ≥ 0, trace(T) ≤ M}

is closed in the weak operator topology, the assertion is evident. Finally, using the fact that A_{n_{k}} − λ1 converges in the strong operator topology to A − λ1 and

(A_{n_{k}} − λ1)B_{k}^{−1}(P_{n_{k}} − Q_{k}) = P_{n_{k}} − Q_{k}

for every k, we may take weak limits in the latter formula to obtain

(A − λ1)C = 1 − Q,

as required □

Remarks. The appendix contains an example which shows that the inclusions of Theorem 2.3 can be proper.

One can imagine computing the eigenvalues of A_{n} for every n = 1, 2, …, picking a point λ ∈ Λ, and observing the distribution of eigenvalues in the vicinity of λ as n becomes large. If λ is essential then the number of eigenvalues in any small interval containing λ will increase without limit. If λ is transient, then there are positive integers p ≤ q with the property that eventually you always see at least p points in the interval but never more than q. Moreover, after p and q are appropriately adjusted, then the same behavior will occur in every sufficiently small interval containing λ.

3. Filtrations and their Banach algebras.

Let A be a self adjoint operator on a Hilbert space H. Suppose we have an orthonormal basis for H, indexed either by ℕ or ℤ, and we form the matrix (a_{ij}) of A relative to this basis. If we form a sequence of finite-dimensional compressions of (a_{ij}) then we may compute Λ and Λ_{e} in principle, and the general results of section 2 imply that σ_{e}(A) ⊆ Λ_{e}. In this section we find conditions on the matrix (a_{ij}) which guarantee first, that every point of Λ is either essential or transient and second, that σ_{e}(A) = Λ_{e}. The criterion is that the series ∑_{k} |k|^{1/2}d_{k} should converge, where d_{k} is the sup norm of the kth diagonal of (a_{ij}). We conclude this section with some comments about the nature of transient points.

In order to deal with both unilateral and bilateral orthonormal bases as well as more general situations in which the dimensions of compressions increase in uneven jumps, it is necessary to work with filtrations. We introduce the concept of degree of an operator and a Banach ∗-algebra D(F) of operators associated with a filtration F. It is important for the applications that it should be easy to estimate the norm in D(F) (see Proposition 3.4).

Definition 3.1.

(1) A filtration of H is a sequence F = {H_{1}, H_{2}, … } of finite dimensional subspaces of H such that H_{n} ⊆ H_{n+1} and

\overline{∪_{n}H_{n}} = H.

(2) Let F = {H_{n}} be a filtration of H and let P_{n} be the projection onto H_{n}. The degree of an operator A ∈ B(H) is defined by

deg(A) = sup_{n≥1} rank(P_{n}A − AP_{n}).

Remarks. deg(A) is either a nonnegative integer or +∞. Moreover, we have deg(A^{∗}) = deg(A), deg(λA) = deg(A) for all λ ≠ 0,

deg(A + B) ≤ deg(A) + deg(B), deg(AB) ≤ deg(A) + deg(B).

Only the last of the four properties is not quite obvious; it follows from the fact that the map A → [P_{n}, A] = P_{n}A−AP_{n} is a derivation, [P_{n}, AB] = [P_{n}, A]B +A[P_{n}, B], which implies

rank[P_{n}, AB] ≤ rank[P_{n}, A] + rank[P_{n}, B].

We conclude from these observations that the set of all finite-degree operators in B(H) is a self-adjoint unital subalgebra of B(H).

Remark. Notice that for any operator A and any projection P we have

P A − AP = P A(1 − P) − (1 − P)AP,

hence

rank(P A − AP) = rank(P A(1 − P)) + rank((1 − P)AP)
= rank((1 − P)A^{∗}P) + rank((1 − P)AP).

It follows that deg(A) < ∞ iff both operators B = A and B = A^{∗} satisfy the condition

sup_{n≥1} rank((1 − P_{n})BP_{n}) < +∞.

Thus, operators of finite degree are abstractions of band-limited matrices.

We associate a Banach ∗-algebra D(F) to a filtration F in the following way. Let D(F) denote the set of all operators A ∈ B(H) having a decomposition into an infinite sum of finite degree operators A_{k}

(3.2)  A = ∑_{k=1}^{∞} A_{k}

in such a way that the sum

s = ∑_{k=1}^{∞}(1 + deg(A_{k})^{1/2})∥A_{k}∥

is finite. Notice that finiteness of s ensures that the series 3.2 converges absolutely with respect to the operator norm. We define |A|_{F} to be the infimum of all such sums s which arise from representations of A as in 3.2.

Proposition 3.3. With respect to the norm | · |_{F} and the operator adjoint, D(F) is a unital Banach ∗-algebra in which |1|_{F} = 1.

proof. Let D be the ∗-algebra of all finite degree operators in B(H), and consider the function ϕ : D → ℝ^{+} defined by

ϕ(A) = (1 + (degA)^{1/2})∥A∥.

The norm on D(F) is defined by

|A|_{F} = inf ∑_{k=1}^{∞} ϕ(A_{k}),

the infimum extended over all sequences A_{1}, A_{2}, … in D satisfying

A = ∑_{k=1}^{∞} A_{k}

together with ∑_{k} ϕ(A_{k}) < +∞. The fact that (D(F), | · |_{F}) is a Banach ∗-algebra follows from straightforward applications of the following properties of ϕ:

(1) ϕ(A) ≥ ∥A∥.

(2) ϕ(λA) = |λ| · ∥A∥, λ ∈ ℂ.

(3) ϕ(A^{∗}) = ϕ(A).

(4) ϕ(AB) ≤ ϕ(A)ϕ(B). For example, (4) follows from the fact that the weight sequence w_{0}, w_{1}, w_{2}, … defined by w_{k} = 1 + k^{1/2} satisfies

w_{k+j} ≤ w_{k}w_{j}, k, j ≥ 0,

together with

deg(AB) ≤ degA + degB.

The identity of B(H) is of degree 0, hence |1|_{F} ≤ 1, while the inequality |1|_{F} ≥ 1 is valid for the unit of any Banach algebra □

D(F) is certainly not a C^{∗}-algebra, but it is dense in B(H) in the strong operator topology. The following gives a concrete description of operators in D(F) for one of the two most important filtrations.

Proposition 3.4. Let {e_{n} : n ∈ ℤ} be a bilateral orthonormal basis for a Hilbert space H and let F = {H_{1}, H_{2}, … } be the filtration defined by

H_{n} = [e_{−n}, e_{−n+1}, …, e_{n}].

Let (a_{ij}) be the matrix of an operator A ∈ B(H) relative to {e_{n}}, and for every k ∈ ℤ let

d_{k} = sup_{i∈ℤ} |a_{i+k,i}|

be the sup norm of the kth diagonal of (a_{ij}). Then

(3.5)  |A|_{F} ≤ ∑_{k=−∞}^{+∞}(1 + |2k|^{1/2})d_{k}.

In particular, A will belong to D(F) whenever the series ∑_{k} |k|^{1/2}d_{k} converges.

proof. We may assume that the sum on the right side of 3.5 converges. Let D_{k} be the operator whose matrix agrees with (a_{ij}) along the kth diagonal and is zero elsewhere. Notice that ||D_{k}|| = d_{k}, and hence

A = ∑_{k=−∞}^{+∞} D_{k},

the sum on the right converging absolutely in the operator norm because

∑_{k} |k|^{1/2}d_{k} < ∞.

We claim that degD_{k} ≤ 2|k| for every k ∈ ℤ. To see that, fix n = 1, 2, … and let P_{n} be the projection onto H_{n}. The operator D_{k} maps the range of P_{n} into the range of P_{n+|k|}, and a careful inspection of the two cases k > 0 and k < 0 shows that the dimension of (1 − P_{n})P_{n+|k|} is at most |k| for every n ≥ 1. It follows that the rank of either operator (1 − P_{n})D_{k}P_{n} or (1 − P_{n})D_{k}^{∗}P_{n} is at most |k|, independently of n. Thus from a previous remark we conclude that degD_{k} ≤ 2|k|.

Now ∥D_{k}∥ = d_{k}, and since the series ∑_{k} |k|^{1/2}d_{k} converges we must have ∑_{k} ∥D_{k}∥ < ∞. Hence

A = ∑_{k=−∞}^{+∞} D_{k}

is an absolutely convergent sum of operators D_{k} satisfying degD_{k} ≤ 2|k|, from which 3.5 follows □

In particular, any operator whose matrix (a_{ij}) is band-limited in the sense that a_{ij} = 0 whenever |i − j| is sufficiently large, must belong to D(F).

The following estimate occupies a key position, and it explains why the norm on D(F) is defined as it is. For B ∈ B(H), we write ∥B∥_{2} = (trace(B^{∗}B))^{1/2} for the Hilbert-Schmidt norm of B.

Lemma 3.6. Let {H_{1}, H_{2}, … } be a filtration of H and let P_{n} be the projection onto H_{n}. Then for every A ∈ D(F) we have

sup_{n≥1} ∥AP_{n} − P_{n}AP_{n}∥_{2} ≤ |A|_{F} < ∞.

proof. We claim first that for every A ∈ B(H) and every n = 1, 2, … we have

(3.7)  trace|AP_{n} − P_{n}AP_{n}|^{2} ≤ deg(A)∥A∥^{2},

where for an operator B ∈ B(H), |B| denotes the positive square root of B^{∗}B. Indeed, since trace|B|^{2} ≤ rank(B)∥B∥^{2}, we can estimate the left side of 3.7 as follows

trace|(1 − P_{n})AP_{n}|^{2} ≤ rank((1 − P_{n})AP_{n}))∥(1 − P_{n})AP_{n}∥^{2}
= rank((1 − P_{n})(AP_{n} − P_{n}A))∥(1 − P_{n})AP_{n}∥^{2}
≤ rank(AP_{n} − P_{n}A)∥A∥^{2} ≤ deg(A)∥A∥^{2}.

By taking the square root of both sides of 3.7 we obtain

∥AP_{n} − P_{n}AP_{n}∥_{2} ≤ deg(A)^{1/2}∥A∥.

Now suppose A ∈ D(F), and choose finite degree operators A_{k} such that

∑_{k}(1 + deg(A_{k})^{1/2})∥A_{k}∥ < ∞

and A = A_{1} + A_{2} + . . .. Using the triangle inequality for the Hilbert-Schmidt norm we see from 3.7 that

∥P_{n}A − P_{n}AP_{n}∥_{2} ≤ ∑_{k=1}^{∞} deg(A_{k})^{1/2}∥A_{k}∥

≤ ∑_{k=1}^{∞}(1 + deg(A_{k})^{1/2})∥A_{k}∥.

The desired inequality follows by taking the supremum over n and the infimum over all such sequences A_{1}, A_{2}, … □

Remark. Notice that Lemma 3.6 implies that for every A ∈ D(F) we have the following uniform estimate of commutator 2-norms

sup_{n} ∥AP_{n} − P_{n}A∥_{2} ≤ \sqrt{2}|A|_{F} < +∞.

Now let A be a self-adjoint operator in B(H), let F = {H_{1}, H_{2}, … } be a filtration, and let A_{n} be the compression of A to H_{n}. The following result is our basic description of the spectrum of A in terms of the eigenvalues of {A_{n}}. It asserts that the essential spectrum of A is precisely the set of essential points, and that all of the remaining points of Λ are transient.

Theorem 3.8. Assume that A = A^{∗} belongs to the Banach algebra D(F). Then (i) σ_{e}(A) = Λ_{e}. (ii) Every point of Λ is either transient or essential.

proof. We will prove that every point in the complement of the essential spectrum of A is transient. Notice that both statements (i) and (ii) follow from this. Indeed, because of Theorem 2.3 we know that σ_{e}(A) ⊆ Λ_{e}. On the other hand, since no transient point can be an essential point, the above assertion implies that the complement of σ_{e}(A) is contained in the complement of Λ_{e}; hence (i). Armed with (i), we see that Λ \ Λ_{e} is the same as Λ \ σ_{e}(A), which by the assertion is contained in the transient points of Λ. Thus we obtain (ii).

To prove this assertion, choose a point λ in the complement of σ_{e}(A). In order to show that λ is transient, we claim first that there is a finite-dimensional projection Q and a positive number ϵ such that for every real number μ ∈ [λ − ϵ, λ + ϵ], the operator

A + Q − μ1

is invertible. Indeed, since λ ∉ σ_{e}(A) it follows that either λ is an isolated point of the spectrum of A whose eigenspace is finite dimensional, or else A−λ1 is invertible. In either case

{ξ ∈ H : Aξ = λξ}

is finite dimensional and if Q denotes the projection onto this eigenspace, then Q commutes with A and A + Q − λ1 is an invertible self-adjoint operator. Because the invertible operators are an open set it follows that there is a neighborhood U of λ with the property that A + Q − μ1 is invertible for every μ ∈ U. We can take [λ − ϵ, λ + ϵ] to be an appropriate closed interval about λ which is contained in such a neighborhood.

Note too that by continuity of inversion, there is a positive number M such that

(3.9)  ∥ (A + Q − μ1)^{−1} ∥≤ M

for every μ satisfying |μ − λ| ≤ ϵ.

We will show that if V is the open interval (λ−ϵ, λ+ϵ) and N_{n}(V) is the number of eigenvalues of A_{n} which lie in V, then the sequence N_{n}(V) is bounded. Thus λ must be a transient point, and that will complete the proof. To that end, note first that by Lemma 3.6 there is a positive number N (in fact, N = |A|_{F}^{2} will do) such that

trace|AP_{n} − P_{n}AP_{n}|^{2} ≤ N,

for every n = 1, 2, . . .. Fix n, and let λ_{1}, λ_{2}, …, λ_{p} be the set of all eigenvalues of A_{n} which lie in V, with repetitions according to the multiplicity of the corresponding eigenspace. Actually, we will show that

(3.10)  p ≤ M^{2}N + dim Q.

Since the right side of 3.10 does not depend on n it will follow that λ is transient, completing the proof.

To prove 3.10, we choose for each k = 1, 2, …, p a unit vector ξ_{k} in the domain of A_{n} (i.e., the range of P_{n}) such that A_{n}ξ_{k} = λ_{k}ξ_{k}, and such that {ξ_{1}, …, ξ_{p}} is an orthonormal set. This is possible because the eigenvectors which belong to different eigenvalues of a self-adjoint operator are mutually orthogonal; in the case of a multiple eigenvalue we simply choose an orthonormal basis for that eigenspace.

Note first that for each k between 1 and p,

(3.11)  Aξ_{k} − λ_{k}ξ_{k} = (1 − P_{n})Aξ_{k}.

Indeed, since P_{n}(Aξ_{k} − λ_{k}ξ_{k}) = A_{n}ξ_{k} − λ_{k}ξ_{k} = 0, we can write

Aξ_{k} − λ_{k}ξ_{k} = (1 − P_{n})(Aξ_{k} − λ_{k}ξ_{k}) = (1 − P_{n})Aξ_{k}.

We claim now that for each k between 1 and p, we have the inequality

(3.12)  1 ≤ M^{2} ∥ (1 − P_{n})Aξ_{k} ∥^{2} + < Qξ_{k}, ξ_{k} > .

To see that, write

1− < Qξ_{k}, ξ_{k} > = ∥ (1 − Q)ξ_{k} ∥^{2} ≤ M^{2} ∥ (A + Q − λ_{k})(1 − Q)ξ_{k} ∥^{2}
= M^{2} ∥ (A − λ_{k}1)(1 − Q)ξ_{k} ∥^{2} = ∥ (1 − Q)(Aξ_{k} − λ_{k}ξ_{k}) ∥
= M^{2} ∥ (1 − Q)(1 − P_{n})Aξ_{k} ∥^{2} ≤ M^{2} ∥ (1 − P_{n})Aξ_{k} ∥^{2} .

The first inequality follows from 3.9, the identity (A−λ_{k})(1−Q)ξ_{k} = (1−Q)(Aξ_{k} − λ_{k}ξ_{k}) follows from the fact that Q and A commute, and the last equality and inequality follow respectively from 3.11 and the fact that 1 − Q is a contraction. Summing the inequality 3.12 on k we obtain

p ≤ M^{2} ∑_{k=1}^{p} ∥ (1 − P_{n})Aξ_{k} ∥^{2} + ∑_{k=1}^{p} < Qξ_{k}, ξ_{k} >
≤ M^{2} trace|(1 − P_{n})AP_{n}|^{2} + trace(Q)
≤ M^{2}N + dimQ,

and 3.10 is established □

Remark 3.13. We conclude this section with some remarks about transient points and other spurious eigenvalues. To provide a context for these remarks, let us return to the classical setting of Szegö’s theorem, in which A_{1}, A_{2}, … is a sequence of Toeplitz matrices. Let f be a bounded measurable real-valued function on the interval [−π, π], and consider the Fourier series of f

f(x) ∼ ∑_{n=−∞}^{∞} a_{n}e^{inx}.

For n = 1, 2, … let A_{n} be the Toeplitz matrix

A_{n} = ( \begin{matrix}a_{0} & a_{−1} & … & a_{−n+1} \\ a_{1} & a_{0} & … & a_{−n+2} \\ ⋮ & & ... & ⋮ \\ a_{n−1} & … & & a_{0} \end{matrix} ) .

Let a and b be respectively the essential inf and sup of f over [π, π]. It is known that the eigenvalues of the sequence {A_{n} : n = 1, 2, … } tend to fill out the entire interval [a, b] (see [11, pp 201–202] and [12, Lemma II.1]). In more concrete terms, suppose that λ ∈ (a, b) and that we are looking in a small neighborhood U = (λ − ϵ, λ + ϵ) of λ. Then U ∩ σ(A_{n}) ≠ ∅ for sufficiently large n, regardless of where λ is located in the interval (a, b). The point we want to make is that this can be misleading, and that what one is observing here may have nothing to do with the spectrum of the operator A.

Indeed, for these operators A we have

σ(A) = σ_{e}(A) = R,

where R is the essential range of the function f. Suppose that f is chosen so that its essential range is disconnected. If we choose a point λ ∈ [a, b] \ R then for every ϵ > 0 we will find the sets of eigenvalues (λ − ϵ, λ + ϵ) ∩ σ(A_{n}) to be permanently nonempty for large n, in spite of the fact that (λ − ϵ, λ + ϵ) contains no points of the spectrum of A if ϵ is small.

Returning to the context of Theorem 3.8, it is not hard to give examples of bandlimited matrices (a_{ij}) which exhibit the same kind of behavior, namely σ(A) = σ_{e}(A) while Λ contains points not in the spectrum of the operator. In this case, 3.8 implies that such points will be transient. More generally, if one starts with a bandlimited operator whose spectrum is properly larger than its essential spectrum, then every point of σ(A) \ σ_{e}(A) will be a transient point (Theorem 3.8). Of course, for such an operator there may also be other transient points which do not belong to σ(A). Unfortunately, we do not know how to distinguish between transient points which belong to the spectrum and transient points which do not. Perhaps it is not even possible to do so. In any case, these observations have led us to the conclusion that transient points should be ignored. In doing that one is also ignoring points of σ(A) \ σ_{e}(A); but since such points are merely isolated eigenvalues having finite multiplicity, the cost is small.

We conclude that if one is interested in approximating spectra of infinite dimensional operators using matrix truncations, then one should not simply look for the eigenvalues of the truncations. Rather, one should weight the count of eigenvalues in a way which eliminates spurious ones occuring in the vicinity of transient points. The simplest example of such weighting is the sequence of ‘densities’

\frac{N_{n}(U)}{n},

where N_{n}(U) is the number of eigenvalues of the n×n truncation of A which belong to the open set U. Indeed, in the present context we obviously have

lim_{n→∞} \frac{N_{n}(U)}{n} = 0

whenever U is a sufficiently small neighborhood of a transient point λ, simply because the sequence N_{1}(U), N_{2}(U), … is bounded. If λ is an essential point on the other hand, then we require conditions under which the limit

lim_{n→∞} \frac{N_{n}(U)}{n}

exists and is positive for every open neighborhood U of λ. The existence of this limit in the case of the multiplication operators A discussed above is a consequence of Szegö’s theorem, as we have pointed out in the introduction. In the following section we take up the asymptotic behavior of these densities in a more general context.

4. Filtrations and convergence of densities.

In this section we shift our point of view somewhat. Rather than consider single self-adjoint operators we consider a concretely presented C^{∗}-algebra of operators A ⊆ B(H) and we introduce the concept of an A-filtration. This is an abstraction of an orthonormal basis for H with respect to which a dense set of the operators in A are band-limited. The point we want to make is that in order to use the methods of this paper to compute the spectrum of a self-adjoint operator in a C^{∗}-algebra C, one should first look for a faithful representation π : C → B(H) with the property that the C^{∗}-algebra A = π(C) admits a natural A-filtration, and then proceed with the analysis associated with Theorem 3.8 or Theorem 4.5 below.

After presenting some examples, we prove a counterpart of Szegö’s theorem which is appropriate for C^{∗}-algebras. In order to keep the statement and proof as simple as possible, we assume that the C^{∗}-algebra has a unique trace. While this is not the most general formulation possible, it does provide exactly the context needed for the applications of the following section. It will be convenient (though not necessary) to assume that C^{∗}-algebras of operators on H contain the identity of B(H).

Definition 4.1. Let A ⊆ B(H) be a C^{∗}-algebra. An A-filtration is a filtration of H with the property that the ∗-subalgebra of all finite degree operators in A is norm-dense in A.

Example 1. The simplest example is that in which A is an AF-algebra having a cyclic vector ξ. If A_{1} ⊆ A_{2} ⊆ … is a sequence of finite dimensional *-subalgebras of A whose union is norm-dense in A, then H_{n} = [A_{n}ξ] defines a filtration for which every element of the union ∪_{n}A_{n} has finite degree.

Example 2. Suppose on the other hand that we have an action of ℤ on a compact Hausdorff space X for which there is a point x_{0} ∈ X whose orbit under ℤ is dense. Then one can write down a representation π of the C^{∗}-algebraic crossed product

C = ℤ × C(X)

and a natural filtration for the C^{∗}-algebra A = π(C). To see that, let {e_{n} : n ∈ ℤ} be an orthonormal basis for a Hilbert space H. The representation π_{0} : C(X) → B(H) and the unitary operator U ∈ B(H) defined by

π_{0}(f)e_{n} = f(n · x_{0})e_{n}
Ue_{n} = e_{n+1}

define a covariant pair for the induced action of ℤ on C(X), and hence the pair (π_{0}, U) defines a representation π of the crossed product. Let

H_{n} = [e_{−n}, …, e_{n}], n ≥ 1,

and let D be the image of C(X) under π_{0}. It is clear that every element of D has degree zero, and that deg(U) = 1. Thus any operator expressible as a finite sum

A = ∑_{k=−n}^{n} D_{k}U^{k}

where D_{k} ∈ D, has finite degree. Since these operators are dense in A it follows that {H_{1} ⊆ H_{2} ⊆ … } is an A-filtration.

Example 3. Let T be the Toeplitz C^{∗}-algebra. We represent T as the C^{∗}-algebra generated by a simple unilateral shift S, acting on an orthonormal basis {e_{1}, e_{2}, … } for H via Se_{n} = e_{n+1}. Let F = {H_{n} : n ≥ 1} be the filtration H_{n} = [e_{1}, e_{2}, …, e_{n}]. Notice that deg(S) = 1. Indeed, letting P_{n} be the projection onto H_{n}, we have

SP_{n} = P_{n}SP_{n} + (P_{n+1} − P_{n})SP_{n},

while P_{n}S = P_{n}SP_{n}. Hence SP_{n} − P_{n}S = (P_{n+1} − P_{n})SP_{n} is a rank-one operator for every n. It follows that the ∗-algebra T_{0} generated by S consists of finite degree operators, and T_{0} is dense in T . Thus F is a T -filtration.

We require the following estimate.

Proposition 4.2. Let F = {H_{n}} be a filtration of H, let P_{n} be the projection onto H_{n}, and let A_{1}, A_{2}, …, A_{p} be a finite set of operators in B(H). Then for every n = 1, 2, … we have

trace|P_{n}A_{1}A_{2} … A_{p}P_{n} − P_{n}A_{1}P_{n}A_{2}P_{n} … P_{n}A_{p}P_{n}| ≤ ∥A_{1}∥ … ∥A_{p}∥ ∑_{k=1}^{p} degA_{k}.

proof. Without loss of generality, we can assume that each operator A_{k} is a contraction and has finite degree. Let us fix n, and define

Δ_{p} = P_{n}A_{1}A_{2} … A_{p}P_{n} − P_{n}A_{1}P_{n}A_{2}P_{n} … P_{n}A_{p}P_{n}.

We claim first that there are contractions B_{1}, B_{2}, …, B_{p}, C_{1}, C_{2}, …, C_{p} such that

Δ_{p} = ∑_{k=1}^{p} B_{k}(1 − P_{n})A_{k}P_{n}C_{k}.

This is easily seen by induction on p. It is trivial for p = 1 (take B = C = 0). Assuming it valid for p, we have

Δ_{p+1} = P_{1}A_{1} … A_{p}(1 − P_{n})A_{p+1}P_{n} + Δ_{p}A_{p+1}P_{n},

and the conclusion is evident from the induction hypothesis on Δ_{p}.

Each operator (1 − P_{n})A_{k}P_{n} is of finite rank, since

rank((1 − P_{n})A_{p}P_{n}) = rank((1 − P_{n})(A_{p}P_{n} − P_{n}A_{p}))
≤ rank(A_{p}P_{n} − P_{n}A_{p}) ≤ degA_{p}.

Thus we may write

trace|Δ_{p}| ≤ ∑_{k=1}^{p} trace|B_{k}(1 − P_{n})A_{k}P_{n}C_{k}|

≤ ∑_{k=1}^{p} ∥B_{k}∥∥C_{k}∥ trace|(1 − P_{n})A_{k}P_{n}| (4.3)

≤ ∑_{k=1}^{p} trace|(1 − P_{n})A_{k}P_{n}|,

where we have used the elementary fact that for any finite rank operator F,

trace|BF C| ≤ ∥B∥∥C∥ trace|F |.

Moreover, for such an F we have

trace|F | ≤ ∥F ∥ · rank(F).

Hence

trace|(1 − P_{n})A_{k}P_{n}| ≤ rank((1 − P_{n})A_{k}P_{n}) = rank((1 − P_{n})(A_{k}P_{n} − P_{n}A_{k}))
≤ rank(A_{k}P_{n} − P_{n}A_{k}) ≤ degA_{k}.

Utilizing the latter inequality in (4.3) we obtain

trace|Δ_{p}| ≤ ∑_{k=1}^{p} degA_{k},

as required. □

Let A be a concrete C^{∗}-algebra. A tracial state of A is a positive linear functional ρ satisfying ρ(1) = 1 and ρ(AB) = ρ(BA) for every A, B ∈ A. One might try to construct tracial states of A by using a filtration F = {H_{n} : n ≥ 1} for H in the following way. Letting P_{n} be the projection on H_{n}, we can define a state ρ_{n} of B(H) by way of

ρ_{n}(T) = \frac{1}{dim(H_{n})}trace(P_{n}T).

The restriction of ρ_{n} to P_{n}B(H)P_{n} is a trace. Of course, ρ_{n} does not restrict to a trace of A, but one might attempt to obtain a trace by taking weak*-limits of subsequences of {ρ_{n} ↾_{A}: n ≥ 1}, or their averages. In general, this program will fail. It will certainly fail if A is isomorphic to a Cuntz algebra O_{n}, n = 2, 3, …, ∞, since O_{n} has no tracial states whatsoever. On the other hand, the following result shows that this procedure will succeed when F is an A-filtration.

Proposition 4.4. Suppose that {H_{1} ⊆ H_{2} ⊆ … } is an A-filtration. For every n = 1, 2, … put d_{n} = dimH_{n}, and let ρ_{n} be the state of A defined by

ρ_{n}(A) = \frac{1}{d_{n}} trace(P_{n}A).

Let R_{n} be the weak^{∗}-closed convex hull of the set {ρ_{n}, ρ_{n+1}, ρ_{n+2}, … }. Then ∩_{n}R_{n} is a nonempty set of tracial states. In particular, A has at least one tracial state.

proof. R_{1}, R_{2}, … is a decreasing sequence of nonvoid compact convex subsets of the state space of A, and hence the intersection R_{∞} is a nonvoid compact convex set of states. We have to show that every element σ ∈ R_{∞} is a trace. Since the finite degree elements are dense in A and σ is bounded, it suffices to show that

σ(AB) = σ(BA)

for operators A, B ∈ A having finite degree. Because of the inequality

|σ(AB) − σ(BA)| ≤ lim sup_{n→∞} |ρ_{n}(AB) − ρ_{n}(BA)|,

it suffices to show that |ρ_{n}(AB) − ρ_{n}(BA)| → 0 as n → ∞. But since

trace(P_{n}AP_{n}BP_{n}) = trace(P_{n}BP_{n}AP_{n})

for every n, we may use the case p = 2 of Proposition 4.2 as follows

|ρ_{n}(AB) − ρ_{n}(BA)| = \frac{1}{d_{n}} |(trace(P_{n}ABP_{n}) − trace(P_{n}BAP_{n})|

≤ \frac{1}{d_{n}} (|trace(P_{n}ABP_{n} − P_{n}AP_{n}BP_{n})| + |trace(P_{n}BAP_{n} − P_{n}BP_{n}AP_{n})|)

≤ \frac{2}{d_{n}} ∥A∥∥B∥(degA + degB).

The right side obviously tends to 0 as n → ∞ □

Remarks. Given a C^{∗}-algebra A ⊆ B(H), it is reasonable to ask if it is always possible to find an A-filtration. Proposition 4.4 implies that if an A-filtration exists then A must have at least one tracial state. Thus, many concrete C^{∗}-algebras cannot be associated with a filtration. For example, if A is isomorphic to one of the Cuntz algebras O_{n}, n = 2, 3, …, ∞, then A has no tracial states whatsoever, and hence A-filtrations do not exist.

We assume throughout the remainder of this section that A has a unique tracial state τ. Then every self-adjoint operator A ∈ A determines a natural probability measure μ_{A} on ℝ by way of

∫_{−∞}^{+∞} f(x) dμ_{A}(x) = τ(f(A))

for every f ∈ C_{0}(ℝ). Clearly the closed support of μ_{A} is contained in the spectrum of A, and if τ is a faithful trace then we have

support(μ_{A}) = σ(A).

Here is a more explicit description of μ_{A}. The GNS construction applied to τ gives rise to a representation of A on another Hilbert space in which the image of A generates a finite factor M. The spectral measure E_{A} of A takes values in the projections of M. Letting tr denote the natural normalized trace on M we have

μ_{A}(S) = tr(E_{A}(S))

for every Borel set S ⊆ ℝ. The measure μ_{A} will be called the spectral distribution of A.

Assuming now that F = {H_{n}} is an A-filtration, we have Theorem 4.5. Let μ_{A} be the spectral distribution of a self-adjoint operator A ∈ A, and let [a, b] be the smallest closed interval containing σ(A). For each n, let d_{n} = dim H_{n} and let λ_{1}, λ_{2}, …, λ_{d_{n}} be the eigenvalues of A_{n} = P_{n}A ↾ H_{n}, repeated according to multiplicity. Then for every f ∈ C[a, b],

lim_{n→∞} \frac{1}{d_{n}} (f(λ_{1}) + f(λ_{2}) + ⋯ + f(λ_{d_{n}})) = ∫_{a}^{b} f(x) dμ_{A}(x).

proof. Let τ_{n} be the state of B(H) defined by

τ_{n}(T) = \frac{1}{d_{n}} trace(P_{n}T).

Noting that τ_{n} restricts to the normalized trace on P_{n}B(H)P_{n}, Theorem 4.5 will become evident once we have established the following two assertions.

(i) For every f ∈ C[a, b], we have

lim_{n→∞} |τ_{n}(f(A)) − τ_{n}(f(P_{n}AP_{n}))| = 0.

(ii) For every B ∈ A,

lim_{n→∞} τ_{n}(B) = τ(B),

where τ is the tracial state of A. To prove (i), notice that since the sequence of linear functionals

f ∈ C[a, b] ↦ τ_{n}(f(A)) − τ_{n}(f(P_{n}AP_{n}))

is uniformly bounded (an upper bound for their norms is 2), it is enough to prove (i) for polynomials f; and by linearity, we may further reduce to the case where f is a monomial, f(x) = x^{p} for some p = 1, 2, . . .. Finally, since the sequence of p-linear forms

B_{n}(T_{1}, T_{2}, …, T_{p}) = τ_{n}(T_{1}T_{2} … T_{p}) − τ_{n}(P_{n}T_{1}P_{n}T_{2}P_{n} … P_{n}T_{p}P_{n})

is uniformly bounded (again, ∥B_{n}∥ ≤ 2), it is enough to prove that

lim_{n→∞} |τ_{n}(A^{p}) − τ_{n}((P_{n}AP_{n})^{p})| = 0

for operators A having finite degree. But in this case, Proposition 4.2 implies that

|τ_{n}(A^{p}) − τ_{n}((P_{n}AP_{n})^{p})| ≤ \frac{p}{d_{n}} ∥A∥^{p}deg(A),

and the right side obviously tends to 0 as n → ∞.

Assertion (ii) follows immediately from Proposition 4.4. For if R_{n} denotes the sequence of convex sets of 4.5, then their intersection must consist of the singleton {τ}; hence the sequence of states

B ∈ A ↦ τ_{n}(B) = \frac{1}{d_{n}} trace(P_{n}B)

has a unique weak^{∗} cluster point τ, hence it must actually converge to τ in the weak^{∗} topology □

Remark 4.6. Let A = A^{∗} ∈ A. If one drops the hypothesis that A has a unique trace, then the averages of the eigenvalues of P_{n}A ↾ H_{n} may fail to converge. However, one can push the method of proof to obtain other useful conclusions in certain cases. For example, if we assume that there are sufficiently many traces in the sense that for every nonzero B ∈ A there is a tracial state τ such that τ(B^{∗}B) > 0, then one can obtain the following information about the density of eigenvalues in the vicinity of a point of the spectrum of A. Let N_{n}(U) denote the number of eigenvalues of P_{n}A ↾ H_{n} which belong to the open set U, and let λ be a point in the spectrum of A. Then for every neighborhood U of λ there is a positive number ϵ such that

\frac{N_{n}(U)}{dim(H_{n})} ≥ ϵ,

for all sufficiently large n. We omit the details since this result is not relevant to our needs below.

5. Applications.

Most one-dimensional quantum mechanical systems are described by a Hamiltonian which is a densely defined unbounded self-adjoint operator on L^{2}(ℝ), having the form

Hf(x) = −\frac{1}{2}f′′(x) + V (x)f(x)

V : ℝ → ℝ being a continuous function representing the potential. If one wishes to simulate the behavior of such a quantum system on a computer, one first has to discretize this differential operator in an appropriate way. Secondly, one has to develop effective methods for computing with the discretized Hamiltonian. For our purposes, we interpret the second goal to be that of computing the spectrum.

In [1], [2], we discussed the problem of discretizing the above Hamiltonians in such a way as to preserve the uncertainty principle. We argued that the appropriate discretization is a bounded self-adjoint operator of the form

(5.1)  H_{σ} = −\frac{1}{2}P_{σ}^{2} + V (Q_{σ})

where P_{σ} and Q_{σ} are the bounded self-adjoint operators given by

P_{σ}f(x) = \frac{1}{2iσ} (f(x + σ) − f(x − σ)),
Q_{σ}f(x) = \frac{1}{σ} sin(σx)f(x).

Here, σ is a small (positive rational) number representing the numerical step size.

In even the simplest cases, one cannot carry out explicit calculations of the spectrum of the discretized Hamiltonians 5.1 (see [3],[4],[5],[7],[9]). On the other hand, one can apply the methods of the prceding sections in a straightforward manner to obtain the spectrum of H_{σ} as the limiting case of finite dimensional eigenvalue distributions.

To see this, let U and V be the unitary operators

Uf(x) = e^{iσx}f(x),
V f(x) = f(x + 2σ).

Then 5.1 can be written in the form H_{σ} = αA+β1, where α and β are real numbers and A is a bounded self-adjoint operator of the form

(5.2)  A = V + V^{∗} + v(\frac{1}{2i}(U − U^{∗})).

Here, v : ℝ → ℝ is an appropriately rescaled version of V. Thus, we are interested in the spectrum of A. Of course, A belongs to the C^{∗}-algebra C^{∗}(U, V) generated by U and V. Following the program outlined in section 4, we will first find a faithful representation of C^{∗}(U, V) in which there is a compatible filtration, and then we will look at the eigenvalues fo the approximating sequence of finite matrices. Since U and V obey the commutation relation

(5.3)  V U = e^{2iσ^{2}}UV

and since σ is a rational number, it follows that C^{∗}(U, V) is an irrational rotation C^{∗}-algebra. Thus in order to define a faithful representation, it is enough to specify a pair of unitary operators U_{1} and V_{1} which satisfy the same commutation relation 5.3. For that, consider a Hilbert space spanned by an orthonormal basis {e_{n} : n ∈ ℤ} and define U_{1} and V_{1} by

U_{1}e_{n} = e^{−2inσ^{2}}e_{n},
V_{1}e_{n} = e_{n+1}.

The pair {U_{1}, V_{1}} is irreducible, it obeys the required commutation relations, and A becomes the tridiagonal operator V_{1} + V_{1}^{∗} + D, where D is the diagonal operator

De_{n} = v(− sin(2σ^{2}n))e_{n}, n ∈ ℤ.

Let A = C^{∗}(U_{1}, V_{1}). Then the sequence of subspaces H_{1}, H_{2}, … defined by H_{n} = [e_{−n}, e_{−n+1}, …, e_{n}] is an A-filtration (see example 2 of section 4). Since concretely represented irrational rotation C^{∗}-algebras never contain nonzero compact operators, the spectrum of A coincides with the essential spectrum of A. Moreover, by Theorem 3.8, the spectrum of A is the set of essential points associated with the sequence of matrices {A_{n} : n ≥ 1}, A_{n} being the compression of A to H_{n}. The remaining points of Λ are all transient points. Finally, since irrational rotation C^{∗}-algebras are simple and have a unique trace, we may conclude from Theorem 4.5 that for every f ∈ C_{0}(ℝ) one has the following convergence of the densities of eigenvalues,

(5.1)  lim_{n→∞} \frac{1}{2n + 1}(f(λ_{1}) + ⋯ + f(λ_{2n+1}) = ∫_{−∞}^{+∞} f(x) dμ_{A}(x),

wehre μ_{A} is the spectral distribution of A and {λ_{1}, …, λ_{2n+1}} is the eigenvalue list of A_{n}.

We remark that this representation is particularly convenient for numerical eigenvalue computations. Indeed, since the matrices A_{1}, A_{2}, … are already in tridiagonal form relative to the obvious basis, there are extremely efficient algorithms for computing the number of eigenvalues of A_{n} which are smaller than a fixed real number.

6. Appendix. Failure of σ_{e}(A) = Λ_{e}. In this section we give an example of a self-adjoint operator A on a Hilbert space spanned by an orthonormal basis {e_{1}, e_{2}, … }, such that the relation of A to the filtration H_{n} = [e_{1}, e_{2}, …, e_{m}], n = 1, 2, … is pathological. We show that if A_{n} is the compression of A to H_{n}, then Λ_{e} contains points not in the spectrum of A, and in particular Λ_{e} ≠ σ_{e}(A).

Specifically, A is a reflection (i.e., a self-adjoint unitary operator) whose essential spectrum is {−1, +1}, whereas 0 ∈ Λ_{e}. In the example H = l^{2}(ℕ), e_{n} is the unit coordinate funtion at n, and A will be the operator in l^{2}(ℕ) induced by a permutation π of ℕ satisfying π^{2} = id. Let I_{n} denote the interval {1, 2, …, n}, and let #S denote the number of elements of a set S. Suppose further that the permutation π has the following property

(A.1)  lim_{n→∞} #(π(I_{n}) \ I_{n}) = +∞.

For each n ≥ 1, let A_{n} be the compression of the operator Aξ(k) = ξ(π(k)) to H_{n} = l^{2}(I_{n}). Since A maps e_{k} to e_{π(k)}, it follows that for every k in the set S_{n} = {k ∈ I_{n} : π(k) ∉ I_{n}}, we have A_{n}e_{k} = 0. Since the cardinality of S_{n} tends to infinity, it follows that 0 is an eigenvalue of every A_{n} and that the multiplicity of this zero eigenvalue tends to infinity. Hence, 0 must belong to Λ_{e}.

Thus it suffices to exhibit an order two permutation having the property A.1. Let E and O denote respectively the set of even and odd integers in ℕ. In order to define a permutation π with π^{2} = id, it is enough to specify a bijection f : E → O; once we have f then we can define π by

π(k) = \{ \begin{matrix}f(k), k ∈ E \\ f^{−1}(k), k ∈ O. \end{matrix}

We will define a bijection f : E → O having the following property with respect to the initial intervals I_{n}:

(A.2)  lim_{n→∞} #(f(E ∩ I_{n}) \ I_{n}) = +∞.

The resulting permutation will have the desired property A.1.

To construct f, split E into two disjoint subsets E_{1} ∪ E_{2}, where E_{1} = 4ℕ is the set of multiples of 4 and E_{2} is the rest. Both E_{1} and E_{2} are infinite of course. First define f on E_{1} by

(A.3)  f(k) = k^{2} + 1.

Let O_{1} = f(E_{1}), and put O_{2} = O \ O_{1}. Noting that O_{2} is infinite (everything in O_{1} is congruent to 1 mod 16, hence O_{2} contains all the other odd numbers), we can complete the definition of f by choosing it to be any bijection of E_{2} onto O_{2}.

To prove A.2, it suffices to show that #(f(E_{1} ∩ I_{n}) \ I_{n}) tends to infinity with n. Noting that

E_{1} ∩ I_{n} = {4k : 1 ≤ k ≤ [n/4]},

we see that

f(E_{1} ∩ I_{n}) = {16k^{2} + 1 : 1 ≤ k ≤ [n/4]}.

Thus every element in the subset P_{n} of E_{1} ∩ I_{n} defined by

P_{n} = \{4k : \frac{\sqrt{n}}{4} < k ≤ \frac{n}{4}\}

gets mapped by f into the complement of I_{n}. Hence

#(f(E_{1} ∩ I_{n}) \ I_{n}) ≥ #P_{n} ≥ \frac{n}{4}(1 − n^{−1/2}).

This estimate implies that #(f(E ∩ I_{n}) \ I_{n}) → ∞, and A.2 follows.

Remark. Notice that this misbehavior cannot be corrected by averaging the eigenvalues of A_{n} as we did in section 4. Indeed, if N_{n}(U) denotes the number of eigenvalues of A_{n} which belong to an open set U, then the above estimate shows that for every neighborhood U of 0 we have

lim inf_{n→∞} \frac{N_{n}(U)}{dimH_{n}} ≥ \frac{1}{4} > 0,

in spite of the fact that 0 does not belong to the spectrum of A.

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