Lower bounds for unbounded operators and semigroups

TL;DR

This paper investigates methods to extend unbounded operators and semigroups on Banach and Hilbert spaces to larger spaces with better spectral properties, providing new results and open questions in operator theory.

Contribution

It extends existing results on bounded operators to unbounded operators and semigroups, offering new bounds and extension techniques in Banach and Hilbert spaces.

Findings

Extended results for unbounded operators on Banach spaces.

Provided new bounds for resolvents of extended operators.

Raised open questions on operator extensions and spectral improvements.

Abstract

Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some estimates for the resolvent of $B$, and also to extend bounded operators related to $A$, for example a semigroup generated by $A$. When $X$ is a Hilbert space, one may also want $Y$ to be Hilbert space. Results of this type for bounded operators have been given by Arens, Read, M\"uller and Badea, and we give some extensions of their results to unbounded operators and we raise some open questions. A related problem is to improve properties of a $C_0$-semigroup satisfying lower bounds by extending it to a $C_0$-group on a larger space or by finding left-inverses. Results of this type for Hilbert spaces have been obtained by Louis and Wexler, and by Zwart, and we give some additional results.