Resolvent conditions and growth of powers of operators

Abstract

Following Bermúdez et al. [5], 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, T n = O(n/ √ log n); if T is absolutely Cesàro bounded, T n = O(n 1/2−ε ) for some ε > 0 (which depends on T ); if T is strongly Kreiss bounded, then T n = 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 > 1.2010 Mathematics Subject Classification. Primary: 47A10, 47A35.

Growth rate of eventually positive Kreiss bounded $$C_0$$-semigroups on $$L^p$$ and $$\mathcal {C}(K)$$

“…Proof. The upper bound of (9) follows from Lemma 3.4 of [5] and the lower bound may be proved similarly. Then, (10) follows from the fact that, for…”

“…Proof of Theorem 1.1. Since T is strongly Kreiss bounded, it is known (see pages 1-2 of [5] that there exists C > 0 such that for every N ∈ N and every x ∈ L p (µ),…”

“…When X is a Hilbert space, Cohen et al [5] proved that (3) implies that T N = O((log N ) κ ) and that this logarithmic control is best possible under (2) in the Hilbert setting.…”

Growth rate of positive Kreiss bounded $C_0$-semigroups on $L^p$

“…The following definitions are well known and studied in the discrete case, that is, when working with the powers of a single operator, see for instance [3] and the references therein. We start by giving the appropriate definitions of Kreiss and Cesàro boundedness properties in the setting of C 0 -semigroups.…”

Behavior of Kreiss bounded $C_0$-semigroups on a Hilbert space

“…The proof of the Theorem 1 is based on techniques given in [2]. We first state and prove two lemmas.…”