Summability and duality

TL;DR

This paper explores the convergence of summability methods in Banach spaces and their duals, establishing conditions for weak and weak*-topology convergence, and illustrating these results across various function spaces.

Contribution

It formalizes the duality of summability methods and provides a general limitation theorem, unifying convergence results across diverse function spaces.

Findings

Convergence of summability methods in a Banach space and its dual are equivalent.

Derived necessary conditions for convergence in the original space.

Applied abstract theorems to multiple function spaces, demonstrating broad applicability.

Abstract

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in $X$. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de Branges-Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems.