On convex-cyclic operators

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].

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“…Kreiss bounded, Kreiss bounded and power bounded operators are equal. 6. Any absolutely Cesàro bounded operator is uniformly Kreiss bounded.…”

Section: On Finite-dimensional Hilbert Spaces the Classes Of Uniformmentioning

confidence: 98%

“…This leads to the following definition: Faghih and Hedayatian proved in [14] that m-isometries on a Hilbert space are not weakly hypercyclic. Moreover, m-isometries on a Banach space are not 1-weakly hypercyclic [6]. However, there are isometries that are weakly supercyclic [23] (in particular cyclic).…”

Section: Corollary 34mentioning

confidence: 99%

Cesàro bounded operators in Banach spaces

Bermúdez

Bonilla

Müller

et al. 2020

JAMA

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“…The notion of convex-cyclicity is a relatively young subject of study which was introduced by Rezaei [23] in 2013. A few more studies were made recently in [4,10,17].…”

Section: Convex-cyclic Weighted Composition Operators and Their Adjointsmentioning

confidence: 99%

Convex-Cyclic Weighted Composition Operators and Their Adjoints

Mengestie

2021

Mediterr. J. Math.

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We characterize the convex-cyclic weighted composition operators $$W_{(u,\psi )}$$ W ( u , ψ ) and their adjoints on the Fock space in terms of the derivative powers of $$ \psi $$ ψ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and, hence, with the weak and strong topologies, and answers a question raised by T. Carrol and C. Gilmore in [5].

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“…Faghih and Hedayatian proved in [14] that m-isometries on a Hilbert space are not weakly hypercyclic. Moreover, m-isometries on a Banach space are not 1-weakly hypercyclic [4]. However, there are isometries that are weakly supercyclic [20] (in particular cyclic).…”

Section: Numerically Hypercyclic Properties Of M-isometriesmentioning

confidence: 99%

Ergodic and dynamical properties of m-isometries

Bermúdez

Bonilla

Müller

et al. 2019

Linear Algebra and its Applications

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An example of a weakly ergodic 3-isometry is provided in [3], we give new examples of weakly ergodic 3-isometries and study numerically hypercyclic misometries on finite and infinite dimensional Hilbert spaces. In particular, all weakly ergodic strict 3-isometries on a Hilbert space are weakly numerically hypercyclic. Adjoints of unilateral forward weighted shifts which are strict m-isometries on 2 (N) are shown to be hypercyclic.

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On convex-cyclic operators

Cited by 18 publications

References 21 publications

Cesàro bounded operators in Banach spaces

Cesàro bounded operators in Banach spaces

Convex-Cyclic Weighted Composition Operators and Their Adjoints

Ergodic and dynamical properties of m-isometries

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