Cesàro bounded operators in Banach spaces

Abstract

We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing absolutely Cesàro bounded operators on ℓ p (N), 1 ≤ p < ∞, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement very limited number of known examples (see [24] and [4]). In [4] Aleman and Suciu ask if every uniformly Kreiss bounded operator T on a Banach spaces satisfies that lim n T n n 1 2 . 4. There exist Kreiss bounded operators which are not Cesàro bounded, and conversely [28].

Resolvent conditions and growth of powers of operators

“…The purpose of this paper is to study the connections between different resolvent conditions and Cesàro boundedness conditions, and the growth properties of T n . Our work continues and complements that of Bermúdez et al [5]. For an overview of the results see Subsection 1.4 below.…”

“…The proof that (4) implies (1) is immediate. Gomilko and Zemánek [18] proved that (2) implies (4), hence (5); thus in reflexive spaces (2) implies mean ergodicity, since T n = O( √ n). If T is power-bounded, then (2) holds (in an equivalent norm T is a contraction and C = 1 in (1)).…”

“…If T is power-bounded, then (2) holds (in an equivalent norm T is a contraction and C = 1 in (1)). By Strikwerda and Wade [43, p. 352], (1) does not imply (5). Bermúdez et al [5] proved that if T on a Hilbert space satisfies (4), then T n = o(n), and then T is mean ergodic.…”

“…By Strikwerda and Wade [43, p. 352], (1) does not imply (5). Bermúdez et al [5] proved that if T on a Hilbert space satisfies (4), then T n = o(n), and then T is mean ergodic. In Section 5 we prove that a positive Cesàro bounded operator on a complex Banach lattice is uniformly Kreiss bounded.…”

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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.

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