Abscissas of weak convergence of vector valued Dirichlet series

TL;DR

This paper investigates the relationships between different types of convergence for vector valued Dirichlet series in locally convex spaces, highlighting how space properties influence these convergence behaviors.

Contribution

It provides a comprehensive comparison of convergence abscissas in various topologies and links these to geometric and topological properties of the underlying spaces.

Findings

Convergence abscissas coincide under certain space conditions

Cotype influences convergence properties in Banach spaces

Nuclearity affects weak convergence behaviors in Fréchet spaces

Abstract

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $\sigma(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fr\'echet spaces play a relevant role.