Suppose I have a sequence of normal operators $T_n$ on some Hilbert space (separable say) that converge in the weak operator topology to $T$ and also have the same essential spectrum and spectrum as we vary $n$. Is it true that
$\sigma_{ess}(T)\subset\sigma_{ess}(T_n)$
or
$$\sigma(T)\subset\sigma(T_n)$$
where $\sigma_{ess}$ denotes the essential spectrum (only one reasonable definition for normal operators)? (Is it even true that $T$ is normal?)
What happens in the case that all the $T_n$ are unitary equivalent?