Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.

3$\begingroup$ My guess is that one will need a suitably strong compactness assumption on the operator to do this (probably trace class will be enough). $\endgroup$ – Terry Tao May 5 '15 at 15:10

2$\begingroup$ There is a subquestion of "what is an analogue of the determinant"  are you already familiar with the Fredholm determinant det(I+T) where T is trace class? $\endgroup$ – Yemon Choi May 5 '15 at 15:33

$\begingroup$ Yes, I am aware of both the Fredholm and functional determinant but I am no expert on either. $\endgroup$ – Benjamin May 5 '15 at 16:01
Several such infinitedimensional extensions of the CH theorem, in terms of the functional determinant, are presented in chapter 4 of Extensions of the CayleyHamilton Theorem with Applications to Elliptic Operators and Frames, Alberto Mokak Teguia (2005).
We show that the CayleyHamilton theorem can be extended to selfadjoint traceclass operators and to closed selfadjoint operators with traceclass resolvent over a separable Hilbert space. It is suggested that it may be possible to drop the selfadjoint restriction.
This was intended as a comment but got too long. I guess it might depend on exactly what you want, but it seems like the short answer is "no"...that is, if you're looking for some function $f$ that you can apply to the operator $T$ so that $f(T)=0$. More precisely, one should at least require that $f$ belong to some functional calculus for $T$. Now the RieszDunford functional calculus lets us define $f(T)$ for any $f$ holomorphic near the spectrum of $T$, but as soon as the spectrum of $T$ is infinite the only holomorphic $f$ giving $f(T)=0$ will be identically $0$. On the other hand if $T$ is normal then any continuous $f$ that vanishes on the spectrum will give $f(T)=0$, but there doesn't seem to be a canonical choice. Conceivably there might be something available for restricted classes of $T$ (say, trace class). Could you say more about what sort of application you have in mind?

$\begingroup$ It was deliberately slightly open ended to see how far the concept can go, sorry if this is a little naughty! The application will take a million years to explain and shed no light on anything! I'm not sure if something like this can be applied to my context yet, that's why I left it open. $\endgroup$ – Benjamin May 5 '15 at 15:16

$\begingroup$ Don't you mean "as soon as the continuous part of the spectrum is infinite"? $\endgroup$ – LSpice May 5 '15 at 16:17

2$\begingroup$ @LSpice If it's a bounded operator, as soon as the spectrum is infinite it has a limit point, and then the only function holomorphic on a neighbourhood of $\sigma(T)$ that is $0$ on $\sigma(T)$ is $0$. $\endgroup$ – Robert Israel May 5 '15 at 18:30

$\begingroup$ This does of course show that if $T$ has finite spectrum $\sigma(T) = \{\lambda_1, \dots, \lambda_n\}$ then taking $f(\lambda) = \prod_{i=1}^n (\lambda  \lambda_i)$ will give $f(T) = 0$. So this is sort of a trivial analogue of CayleyHamilton. $\endgroup$ – Nate Eldredge May 5 '15 at 19:01

$\begingroup$ @RobertIsrael, yes, you're right. I was thinking about densely defined operators like ($i$ times) differentiation on $\text L^2$, but I don't know anything about their functional calculus. $\endgroup$ – LSpice May 5 '15 at 19:12
Consider a nonlinear Partial Differential Equation $$\partial_tu=N[u],$$ where $u=u(t,x)$. A travelling wave (TW) is a solution $u(t,x)=U(xct)$ where $c$ is a constant. Up to a change of frame, one may assume that the wave is stationary: $u=U(x)$. When one linearizes about $U$, one obtain an equation $$\partial_tv={\cal L}v,$$ where the differential operator ${\cal L}$ is linear, with variable coefficients. The stability of $U$ can be studied at three level : nonlinear, linear or spectral. The latter is defined in terms of the spectrum of ${\cal L}$ over $L^2({\mathbb R})$.
For several decades, the main strategy adopted to analyze $\sigma({\cal L})$ has been to construct an Evans function $E(z)$. This is a holomorphic function whose domain is (at least) the open set of numbers $z$ for which ${\cal L}z{\bf id}:H^2\rightarrow L^2$ is Fredholm of index zero.
I believe that this is the best that can be done for a fairly large class of interesting operators. The Evans function is only an avatar of the characteristic polynomial. It has three main flaws:
 it is not defined for all $z\in{\mathbb C}$,
 it is highly nonunique,
 there is no counterpart of CayleyHamilton.
To get around some of the problems Mike Jury mentioned, we can try something like this.
Suppose $T$ is a compact normal operator, with eigenvalues $\lambda_n$, $n=1,2,\ldots$ (in particular $\lambda_n \to 0$ as $n \to \infty$), and all $\lambda_n$ are contained in the disk
$D_1 = \{z: 1+z < 1\}$ (in particular $0$ is not an eigenvalue), and
$\sum_n 1  1 + \lambda_n < \infty$. Let $\mu_n = 1 + \lambda_n$. Then we can form the Blaschke product
$$ B(z) = \prod_{n=1}^\infty \dfrac{\mu_n (\mu_n  z)}{\mu_n (1  \overline{\mu_n} z)} $$
(where the factor is $z$ if $\mu_n = 0$). Then $B(z)$ is a bounded
analytic function on the unit disk, with zeros exactly at $\mu_n$. We conclude that $B(I+T) = 0$ (where this is defined by the functional calculus on bounded Borel functions for normal operators).
Since none of the above responses mention the seminal work of Foias and Sz.Nagy on contractions on Hilbert space which is clearly motivated by the desire to extend the circle of ideas around the Cayley Hamilton theorem to operators on Hilbert space, I would like to add a reference.
As I see it, the basic ideas of the finite dimensional theory are: A functional calculus (over the polynomials); the existence of a polynomial which annihilates a given matrix; the existence of a minimal such polynomial (follows from the above and the fact that ideals in the ring of the polynomials are principal).
These are carried over to the infinite dimensional case as follows: establishment of an $H^\infty$functional calculus for a suitable class of contractions (completely nonunitary). The family of $C_0$contractions is then defined to consist of those which are annihilated by some nonvanishing function and examples and characterisations are given. Finally the existence of a minimal such inner function is deduced as before from the fact that every closed ideal of $H^{\infty}$ is principal. An interesting twist is that one has to work with a weaker topology than the normthe socalled strict topology (Buck, Rubel).
A succinct introduction can be found in the wiki article "Contractions (operator theory)" and a complete account in "Harmonic Analysis of Operators in Hilbert Space" by Sz.Nagy et al.