Resolvent conditions and growth of powers of operators

TL;DR

This paper investigates the growth rates of operator powers under various resolvent conditions, establishing new bounds and ergodic properties, especially in Hilbert and reflexive spaces.

Contribution

It provides new characterizations of power-boundedness and growth rates of operators under resolvent conditions, extending previous results with precise bounds.

Findings

Power-boundedness characterized by Cesàro boundedness of both T and T*

Operators satisfying Kreiss condition have growth rate O(n/√log n)

Kreiss bounded operators on reflexive spaces are Abel ergodic

Abstract

Following Berm\'udez et al. (ArXiv: 1706.03638v1), we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Ces\`aro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^*$ are absolutely Ces\`aro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $\|T^n\|=O(n/\sqrt {\log n})$; if $T$ is absolutely Ces\`aro bounded, $\|T^n\|=O(n^{1/2 -\varepsilon})$ for some $\varepsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $\|T^n\|=O((\log n)^\kappa)$ for some $\kappa >0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Ces\`aro means of order $\alpha$ converge strongly when $\alpha >1$.