Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators
Let \(X\) be a complex Banach space and let \(L(X)\) denote the space of bounded operators on \(X\). An operator \(T \in L(X)\) is said to have Bishop's property \((\beta)\) if, whenever \((f_{n})_{n}\) is a sequence of analytic \(X\)-valued functions such that \((\lambda - T)f_{n}(\lambda) \rightarrow 0\) uniformly on the compact subsets of an open subset \(U\) of the complex plane, then \(f_{n}(\lambda) \rightarrow 0\) uniformly on the compact subsets of \(U\). Among other results, the authors show that for several classes of analytic symbols \(\phi\), the corresponding generalized Cesàro operator \(S_{\phi}\) acting on the Hardy space \(H^{p}, 1 < p < \infty\), has Bishop's property \((\beta)\) and admits a decomposable extension.
scientific article; zbMATH DE number 2131157
scientific article