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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?

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    $\begingroup$ en.wikipedia.org/wiki/Szeg%C5%91_limit_theorems $\endgroup$
    – Steve Huntsman
    Commented Mar 5, 2019 at 13:32
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    $\begingroup$ @SteveHuntsman: The relevance of this link to the question isn't clear to me, can you please elaborate. $\endgroup$
    – Christian Remling
    Commented Mar 5, 2019 at 15:38
  • $\begingroup$ For Toeplitz matrices the Szego theorems show how the spectrum converges to what electrical engineers call the transfer function of the corresponding filter. $\endgroup$
    – Steve Huntsman
    Commented Mar 6, 2019 at 1:41
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    $\begingroup$ 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? $\endgroup$
    – David Ketcheson
    Commented Mar 6, 2019 at 4:32
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    $\begingroup$ In any case, perhaps you are looking for something like this. $\endgroup$
    – David Ketcheson
    Commented Mar 6, 2019 at 4:36

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