Divergence Behavior of Sequences of Linear Operators with Applications

Abstract

In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach–Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., \(\ell ^p\), \(1\le p < \infty \), C[0, 1], \(L^p\), \(1< p <\infty \), as well as Paley–Wiener and Bernstein spaces, have the s-property. Finally, consequences for the convergence behavior of sampling series and system approximation processes are shown.