Let $A_n$ be a sequence of bounded (perhaps compact?) operators on a complex Hilbert space, which all have the same spectrum. Suppose the sequence converges in operator norm (or perhaps in a weaker way?) to an operator $A$. Does this imply that $A$ has the same spectrum?
Suppose the spectra of the $A_n$ converge as sets (perhaps monotonely?) to some subset of $\mathbb{C}$. Is this related to the spectrum of the limit?