Abstract:This paper investigates the growth of the sequences $(\|T^n(I-T)^k\|)_{n\ge 1}$ for bounded linear operators on Banach spaces under various resolvent conditions. Focusing on $\alpha$-Ritt operators that are also strongly Kreiss bounded, we show that the growth estimates established by Nevanlinna for Kreiss bounded operators can be significantly refined within the Hilbert space setting, nearly attaining the optimal rates obtained by Seifert for the more restrictive power-bounded case. We further introduce the class of $(\alpha, \beta)$-RK operators as a generalization of both Ritt and Kreiss-type conditions. For these operators, we derive comprehensive growth estimates which, for certain ranges of $\alpha$ and $\beta$, yield improvements over existing bounds in the literature.
Particular attention is given to $\beta$-Kreiss operators, for which we provide a characterization via Cesàro-type means. We show that, in contrast to the well-known case $\beta = 1$, the power growth estimate $\|T^n\| = O(n^{\beta})$ is sharp whenever $\beta > 1$. The optimality of our estimates is discussed in several cases, relying on constructions and techniques developed by Nevanlinna, Spijker, and Borovykh. We conclude by providing a characterization of Ritt operators that appears to be absent from the literature.
From: Loris Arnold [view email]
[v1]
Fri, 12 Jun 2026 10:04:12 UTC (31 KB)
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