Resolvent conditions and growth of powers of operators

2019/12/22 by Guy Cohen, Cohen, Guy, Christophe Cuny +5
Mathematics · #Holomorphic and Operator Theory #Advanced Topics in Algebra #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.1912.10507

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^* are absolutely Cesàro bounded. In Hilbert spaces, we prove that if T satisfies the Kreiss condition, ‖Tⁿ‖=O(n/√ log n); if T is absolutely Cesàro bounded, ‖Tⁿ‖=O(n^{1/2 -ε}) for some ε >0 (which depends on T); if T is strongly Kreiss bounded, then ‖Tⁿ‖=O((log n)^κ) for some κ>0. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order α converge strongly when α>1.

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