# Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $$T:\ell^2\to\ell^2$$ be a compact linear operator. Let $$[T]=(a_{i,j})_{i,j=1}^{\infty}$$ be the representing infinite matrix of $$T$$ with respect to the canonical base. Let $$T_n$$ be the finite rank operator defined by the matrix $$(a_{i,j})_{i,j=1}^{n}$$ embedded into an infinite matrix. Thus $$T_n\to T$$ in norm.

Can we approximate the eigenvalues of $$T$$ with eigenvalues of $$T_n$$? or more specifically, given $$\lambda$$ an eigenvalue of $$T$$, is there a sequence $$(\lambda_n)$$ such that $$\lambda_n$$ is an eigenvalue of $$T_n$$ and $$\lambda_n\to\lambda$$? Any recommended reference?

• en.wikipedia.org/wiki/Szeg%C5%91_limit_theorems – Steve Huntsman Mar 5 at 13:32
• @SteveHuntsman: The relevance of this link to the question isn't clear to me, can you please elaborate. – Christian Remling Mar 5 at 15:38
• For Toeplitz matrices the Szego theorems show how the spectrum converges to what electrical engineers call the transfer function of the corresponding filter. – Steve Huntsman Mar 6 at 1:41
• Your last sentence is unclear to me. Do you mean to ask "do the eigenvalues of $T_n$ necessarily approximate those of $T$ in some sense?" or "is it possible that the eigenvalues of $T_n$ approximate $T$ (in some sense)?" or something else? – David Ketcheson Mar 6 at 4:32
• In any case, perhaps you are looking for something like this. – David Ketcheson Mar 6 at 4:36