2020
DOI: 10.1016/j.jmaa.2020.124035
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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 ) fo… Show more

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Cited by 8 publications
(6 citation statements)
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“…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…”
Section: Auxilliary Resultsmentioning
confidence: 92%
See 2 more Smart Citations
“…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…”
Section: Auxilliary Resultsmentioning
confidence: 92%
“…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),…”
Section: Auxilliary Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…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.…”
Section: Kreiss and Cesàro Conditions For C 0 -Semigroupsmentioning
confidence: 99%
“…The proof of the Theorem 1 is based on techniques given in [2]. We first state and prove two lemmas.…”
mentioning
confidence: 99%