\sect {\bf The Fredholm Deteminant}
\proclaim Theorem.
Suppose $L$ is a compact operator on a ON Banach module $E$ over
$A$. If $L$ has norm at most
 $ |a|$
where $a\in A$,
 then $P_L(T)$ is
an element of $
A^0[[aT]]$ and is entire in $T$.  Also,
 $P_L(T)$ is characterized by: \pen{
\parindent10pt
\vbox{(i) If $\stt{L_n}_{n\ge 0}$ is
a sequence of
compact operators
on $E$,  and
$L_n\ra L$ then $P_{L_n}\ra P_L$ coefficientwise.}
\ni\vbox{(ii) If the image of $L$ in $E$ 
is contained in a direct factor $F$ of
finite rank over $A$ of $E$ such that
the projection from
$E$ onto $F$ is continuous
then $$P_L(T)=\det(1-TL|{F}).$$}}
In particular, $P_L(T)$  depends only  on the topology.\sk

\pr   I will prove $P_L(T)$ is entire in $T$ and (i). Let $({e_i})_{i\ge 0}$ be an ONB. We can suppose
$|L|\le 1$. Suppose
$L(e_i)=\sum_j n_{i,j}e_j$.  For a finite set $S$ of non-negative integers and a permutation $\sig$ of $S$, set
$$n_{S,\sig}=\prod_{i\in S}n_{i\, \sig(i)}$$
Then
$$P_L(T)=1+c_1T+c_2T^2+\cdots,$$
where
$$ c_m=(-1)^m\sums{S,\sig}_{|S|=m}\eps_\sig n_{S,\sig}.$$


 Now let $R_1\ge R_2\ge\cdots $ be the numbers $\dsp r_j=\sup_{i\ge0}|n_{ij}|$. It follows that 
$$|c_m|\le R_1R_2\cdots R_m,$$
so 
$$|c_m|M^m\le (R_1M)(R_2M)\cdots (R_mM).$$\vsk1

Now suppose, $|L'-L|<\eps<1$.
\vsk1

\beginsection Some other key facts.

\proclaim Remark.  If $L\colon  M\ra N$ is compact and $F\colon N\ra
M$ is continuous, then $L\circ F$ and $F\circ L$ are compact.

\ni(i) If $u$ and $v$ are compact operators on $E$,
$$\det(1-Tu)\det(1-Tv)=\det((1-Tu)(1-Tv)).$$

\ni(ii) Suppose $E_1$ 
and $E_2$ are 
orthonormizable Banach modules over $A$.
  Suppose
$u$ is a compact
homomorphism from $E_1$ to $E_2$
and $v\colon E_2\ra E_1$ is a
continuous homomorphism.  Then
$P_{u\circ v}(T)=P_{v\circ u}(T)$.





\ni (iii) if $\phi\colon A\ra B$ is a homomorphism of Banach algebras then $\phi^*E=:E\otimes_AB$ is orthonormizable over $B$ and 
$$P_{\phi^*L}(T)=\phi(P_L(T)).$$

Given this one can define the characteristic series of a continuous operator $V$ on $M$ if one only asssumes $M$ is ``locally orthonormizable.''\vsk1



\beginsection Riesz Theory

Suppose $u$ is  a compact operator on $E$. Let $A\sstt{T}$ denote the
ring of entire series over $A$.  For a polynomial of degree $d$ whose
leadin coefficient is a unit, $F(T)$, let $F^*(T)=T^dF(T\iv)$.\sk

\proclaim Theorem.   Suppose $P_u(T)=Q(T)S(T)$ where 
$S\in A\sstt{T}$ and 
$Q$ is a polynomial whose leading coeicient is a unit  such that $Q(0)=1$
 and which is relatively prime to
 $S$.
Then there is a unique
 direct sum decomposition
$$E=N_u(Q)\oplus F_u(Q)$$
of $E$ into closed
submodules
 stable by $ u$ such that $N_u(Q)$ is projective of rank $\deg Q$, $Q^*(u)N_u(Q)=0$ and $Q^*(u)$ is 
invertible on $F_u(Q)$.  Moreover, $N_u(Q)$ and $F_u(Q)$ are locally
equivalent to orthonomizable modules and
$P_{u|_{N_u(Q)})}(T)=Q(T)\ \hba\ P_{u|_{F_u(Q)}}(T)=S(T)$.