Browder's theorems through localized SVEP (Q2504845)

A bounded linear operator \(T\in L(X)\) on a Banach space \(X\) is said to satisfy ``Browder's theorem'' if its Browder spectrum \(\sigma_b(T)\) coincides with the Weyl spectrum \(\sigma_w(T)\). \(T\) is said to satisfy ``a-Browder's theorem'' if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum of \(T\). Several conditions equivalent with these properties are given in this paper. Let us mention some of them: \(T\) satisfies Browder's theorem iff \(T^*\) satisfies Browder's theorem iff \(T\) has single valued extension property (SVEP) at every \(\lambda\not\in\sigma_w(T)\) iff \(T^*\) has SVEP at every \(\lambda\not\in\sigma_w(T)\) iff the set of all Riesz points \(p_{00}(T):=\sigma(T)\setminus\sigma_b(T)=\Delta(T)\) (the set of generalized Riesz points of \(T\)). \(T\) satisfies Browder's theorem iff every \(\lambda\in\Delta(T)\) is an isolated point of \(\sigma(T)\) iff \(\Delta(T)\subseteq \partial\sigma(T)\) iff \(\operatorname{int}(\Delta T)=\emptyset\) iff \(\sigma(T)=\sigma_w(T)\cup \operatorname{iso}\sigma(T)\). The concluding section of the paper under review is devoted to operators \(T\) satisfying Weyl's theorem, i.e., for which the set of all generalized Riesz points \(\Delta(T)=\sigma(T)-\sigma_w(T)\). Applying the results of the first part, the authors obtain some conditions equivalent to Weyl's theorem.