Weyl’s Type Theorem and a Local Growth Condition

A bounded linear operator T Telement of L(X) acting on a Banach space satisfies a local growth condition of order m for some positive integer m; T Telement of loc(Gm), if for every closed subset F of the set of complex numbers and every x in the glocal spectral subspace XT (F) there exists an analytic function f : C F→X such that (T -λI) f (λ)≡ x and ǁ f (λ) ǁ ≤ M [dist(λ,F)]^-m ǁ x ǁ for some M 0 (independent of F and x). In this paper, we study the stability of generalized Browder-Weyl theorems under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with T.