Resolvent conditions and growth of powers of operators on $L^p$ spaces

TL;DR

This paper investigates how the growth of operator powers on $L^p$ spaces is influenced by resolvent conditions and boundedness assumptions, emphasizing the roles of type, cotype, and UMD properties.

Contribution

It provides new insights into the growth behavior of operator powers on $L^p$ spaces under specific resolvent and boundedness conditions, utilizing Fourier multiplier techniques.

Findings

Growth rates depend on resolvent conditions and Cesàro boundedness.

UMD property of $L^p$ spaces is crucial for certain proofs.

Results connect operator theory with harmonic analysis on Banach spaces.

Abstract

Let $T$ be a bounded linear operator on $L^p$. We study the rate of growth of the norms of the powers of $T$ under resolvent conditions or Ces\`aro boundedness assumptions. Actually the relevant properties of $L^p$ spaces in our study are their type and cotype, and for $1<p<\infty$, the fact that they are UMD. Some of the proofs make use of Fourier multipliers on Banach spaces, which explains why UMD spaces come into play.