Spectral mapping theorems and perturbation theorems for Browder’s essential spectrum

Abstract:

If $T$ is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of $T,\operatorname {ess} (T)$ [1]. We derive below spectral mapping theorems and perturbation theorems for Browder’s essential spectrum. If $T$ is a bounded linear operator and $f$ is a function analytic on a neighborhood of the spectrum of $T$, we prove that $f(\operatorname {ess} (T)) = \operatorname {ess} (f(T))$. If $T$ is a closed, densely defined linear operator with nonempty resolvent set and $f$ is a polynomial, the same theorem holds. For a closed, densely defined linear operator $T$ and a bounded linear operator $B$ which commutes with $T$, we prove that $\operatorname {ess} (T + B) \subseteq \operatorname {ess} (T) + \operatorname {ess} (B) = \{ \mu + v:\mu \in \operatorname {ess} (T),v \in \operatorname {ess} (B)\}$. By making additional assumptions, we obtain an analogous theorem for $B$ unbounded.