Continuous extension of a densely parameterized semigroup

TL;DR

This paper proves that weakly continuous contractive semigroups on certain Banach spaces, defined over dense sub-semigroups of positive reals, can be extended to the entire positive real line, including non-linear cases.

Contribution

It establishes the extension of densely parametrized semigroups to the full positive real line, including non-linear, non-expansive semigroups, and characterizes extendable semigroups.

Findings

Extension of weakly continuous contractive semigroups to the positive real line.

Extension results for non-linear, non-expansive semigroups.

Characterization of densely parametrized semigroups extendable to the positive reals.

Abstract

Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a weakly continuous semigroup over the positive real numbers. We obtain similar results for non-linear, non-expansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over the positive real numbers.