Degenerate $C$-distribution semigroups in locally convex spaces

TL;DR

This paper explores the theory of degenerate C-distribution semigroups in barreled locally convex spaces, focusing on their generators and regularizing operators, including exponential subclasses, extending existing semigroup theory.

Contribution

It introduces the concept of multivalued generators and non-injective regularizing operators for degenerate C-distribution semigroups in locally convex spaces, with new theoretical insights.

Findings

Characterization of generators as multivalued linear operators

Extension of semigroup theory to non-injective regularizing operators

Analysis of exponential subclasses of degenerate C-distribution semigroups

Abstract

The main purpose of this paper is to investigate degenerate $C$-distribution semigroups in the setting of barreled sequentially complete locally convex spaces. In our approach, the infinitesimal generator of a degenerate $C$-distribution semigroup is a multivalued linear operator and the regularizing operator $C$ is not necessarily injective. We provide a few important theoretical novelties, considering also exponential subclasses of degenerate $C$-distribution semigroups.