Let $A\in \mathbb{C}^{n\times n}$. The Cayley-Hamilton theorem states that if $p(x)$ is the characteristic polynomial of $A$, i.e. $p(\lambda) = \det(\lambda I-A)$, then $A$ satisfies the corresponding matrix polynomial: $p(A)=0$.
I am wondering if the theorem is still true or can be adapted if instead $A$ is an infinite matrix over $\mathbb{C}$, i.e. $A\in \mathbb{C}^{\mathbb{N}\times\mathbb{N}}$. The proofs I have seen involve finite-dimensional vector spaces or the division algorithm, which may not hold in infinite-dimensional spaces. Here we must take $A$ to be trace class; roughly speaking, this asserts that if $(e_k)_k$ is the standard basis, then the sum $\sum_{k=1}^{\infty}\langle |A| e_k,e_k\rangle$ is finite, where $|A|$ is the operator norm . One could define the identity matrix $I$ in an analogous fashion to the finite case and obtain the characteristic function $P$: $$ P(\lambda) = \det(\lambda I-A) $$Here I take $\det$ to be the Fredholm determinant.
Apologies if this post is unclear as I haven't studied functional analysis, so I may be misunderstanding the subtleties involved.