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

Abstract

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