The growth bound for strongly continuous semigroups on Fr\'echet spaces

TL;DR

This paper extends the concepts of growth and spectral bounds for strongly continuous semigroups from Banach spaces to Fréchet spaces, revealing new phenomena and establishing key inequalities.

Contribution

It introduces the notions of growth and spectral bounds for semigroups on Fréchet spaces and demonstrates that Banach space inequalities extend to this broader setting.

Findings

The Banach space inequality $s(A) \\leq \\omega_0(T)$ extends to Fréchet spaces.

Examples show effects on bounds in Fréchet spaces differ from Banach spaces.

New phenomena in bounds are identified in the Fréchet space setting.

Abstract

We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fr\'echet spaces and show that the Banach space inequality $s(A)\leqslant\omega_0(T)$ extends to the new setting. Via a concrete example of an even uniformly continuous semigroup we illustrate that for Fr\'echet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.