Strongly continuous and locally equicontinuous families of operators and their relation to bi-continuity

TL;DR

This paper investigates families of operators that are strongly continuous and locally equicontinuous on certain locally convex spaces, relating these concepts to bi-continuity and generalizing existing results for specific operator classes.

Contribution

It establishes connections between strong continuity, local equicontinuity, and bi-continuity, extending known results to broader classes of operator families.

Findings

Relates bi-continuity to strong and local equicontinuity in Saks spaces.

Generalizes results for bi-continuous semigroups and cosine families.

Provides a unified framework for operator families in locally convex spaces.

Abstract

We study strongly continuous and locally equicontinuous families of operators on sequentially complete Hausdorff locally convex spaces. In case of Saks spaces, we relate the general notions to bi-continuity as well as equitightness. In this way, we recover and also generalise known results for special classes of operator families such as bi-continuous ($C$-)semigroups and ($C$-)cosine families by well-known results for the corresponding families in Hausdorff locally convex spaces.