The Lumer-Phillips Theorem For Two--parameter $C_0$--semigroups

TL;DR

This paper extends the classical Lumer-Phillips theorem to characterize the generators of two-parameter C_0-semigroups of contractions, providing necessary and sufficient resolvent conditions for such operators.

Contribution

It introduces a characterization of infinitesimal generators for two-parameter C_0-semigroups of contractions, generalizing existing one-parameter results.

Findings

Provides resolvent-based necessary and sufficient conditions

Characterizes generators of two-parameter contraction semigroups

Extends classical semigroup theory to multi-parameter setting

Abstract

In this paper we extend the Lumer-Phillips theorem to the context of two--parameter C_0-semigroup of contractions. That is, we characterize the infinitesimal generators of two--parameter C_0-semigroups of contractions. Conditions on the behavior of the resolvent of operators, which are necessary and sufficient for the pair of operators to be the infinitesimal generator of a C_0-semigroup of contractions are given.