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?