Suppose that $F$ and $F_n$ ($n \in \mathbb N$) are linear endomorphisms of a Hilbert space such that
- $F$ is Hilbert-Schmidt (i.e., compact),
- $F_n$ is finite rank for each $n$, and
- $F_n$ converges to $F$ in the Hilbert-Schmidt norm as $n → ∞$.
(I am not assuming self-adjointness anywhere.)
True or false?
The spectrum of $F_n$ converges to that of $F$ (e.g., in Hausdorff distance).
Moreover,
if it is true, can one deduce the (algebraic) multiplicity of each non-zero eigenvalue of $F$?
References most welcome! I know this appears similar to other related questions on convergence of spectra.
(I rooted around in a few FA books for such a result, but the results there focus on self-adjoint operators and don't mention compact operators.)
