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.



Analogue of Cayley Hamilton theorem for operators on Hilbert space
Several such infinite-dimensional extensions of the CH theorem, in terms of the [functional determinant](https://en.wikipedia.org/wiki/Functional_determinant), are presented in chapter 4 of [Extensions of the Cayley-Hamilton Theorem with Applications to Elliptic Operators and Frames](http://dc.etsu.edu/cgi/viewcontent.cgi?article=2181&context=etd), Alberto Mokak Teguia (2005).



> 
> We show that the Cayley-Hamilton theorem can be extended to
>  self-adjoint trace-class operators and to closed self-adjoint
>  operators with trace-class resolvent over a separable Hilbert space. 
>  It is suggested that it may be possible to drop the self-adjoint restriction.
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