Resolvent conditions and growth of powers of operators

Abstract

Following Bermúdez 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 Cesàro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^{\ast }$ are absolutely Cesàro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $\Vert T^n\Vert =O(n/\sqrt{\log n})$; if $T$ is absolutely Cesàro bounded, $\Vert T^n\Vert =O(n^{1/2-\epsilon })$ for some $\epsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $\Vert T^n\Vert =O((\log n{)}^{\kappa })$ for some $\kappa >0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order $\alpha$ converge strongly when $\alpha >1$.

Added references [35] and [38] and updated some remarks. A note regarding one of the problems was added to Section 6