Dunford's property \((C)\) and Weyl's theorems.
A (bounded linear) operator \(T\) on a Banach space \(X\) satisfies Weyl's theorem if its Weyl spectrum coincides with the complement in the spectrum of the isolated point spectrum. An operator \(T\) on \(X\) satisfies the single valued extension property if each complex number has a neighbourhood on which \(f=0\) is the only \(X\)-valued analytic function satisfying \((T-\lambda)f(\lambda)=0\). Dunford's property (C) is similar but stronger. Conditions are given relating property (C) with Weyl's theorem for operators which intertwine in the sense that \(AY=YB\). Weyl's theorem is proved for various classes of operators including \(M\)-hyponormal operators. Moreover, the Weyl spectra of these operators satisfy the spectral mapping theorem with respect to functions analytic in a neighbourhood of the spectrum.
scientific article; zbMATH DE number 1802507
scientific article