PLB-spaces of holomorphic functions with logarithmic growth conditions

TL;DR

This paper investigates the properties of PLB-spaces of holomorphic functions with weights that decay logarithmically, extending previous work on polynomial decay weights to a new class of weights.

Contribution

It provides a characterization of ultrabornological and barrelled properties for these spaces with logarithmic decay weights, expanding understanding of their topological structure.

Findings

Characterization of ultrabornological spaces with logarithmic decay weights

Conditions for barrelledness in these PLB-spaces

Extension of previous polynomial decay results to logarithmic decay

Abstract

Countable projective limits of countable inductive limits, called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet. In a previous article, the author extended their investigation to the case of holomorphic functions and characterized when spaces over the unit disc w.r.t. weights whose decay, roughly speaking, is neither faster nor slower than that of a polynomial are ultrabornological or barrelled. In this note, we prove a similar characterization for the case of weights which tend to zero logarithmically.