Suppose that $T,S$ are densely defined, closed (unbounded) operators on a separable Hilbert space such that there exists $z \in \mathbb C$ in the intersection of resolvent sets of $T$ and $S$ for which $(T-z)^{-1}-(S-z)^{-1}$ is compact. Does it follow that $T$ and $S$ have the same essential spectrum?
Remark: in the application I have in mind $S-T$ is unbounded and domains of $S$ and $T$ are not necessarily equal.
Best Answer
It has been shown in J. J. Buoni ''An Essential Spectra Mapping Theorem'', J. Math. Anal. Appl. 56 (1976) that $$ \sigma_{\mathrm{ess}}(S) = \{ z' \in \mathbb C \, | \, \exists w \in \sigma_{\mathrm{ess}}((S-z)^{-1}) \ \ (z'-z)w =1 \}. $$ The same is true with $T$ replaced by $S$. Since $(T-z)^{-1}-(S-z)^{-1}$ is compact, $\sigma_{\mathrm{ess}}((T-z)^{-1}) = \sigma_{\mathrm{ess}}((S-z)^{-1})$, so the above formula entails $\sigma_{\mathrm{ess}}(S) = \sigma_{\mathrm{ess}}(T)$.