Ergodic and dynamical properties of m-isometries

Bermúdez, Teresa; Bonilla, A.; Müller, Veronika; Peris, Alfredo

doi:10.1016/j.laa.2018.09.022

Cited by 7 publications

(7 citation statements)

References 22 publications

(15 reference statements)

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“…The notion of m-isometric operator was introduced by Agler in [4] and initially investigated by Agler and Stankus in [5]. Recently, there have been published many papers devoted to problems related to m-isometric operators (see [1,6,[10][11][12]18,23]). In particular, much attention was paid to misometric unilateral and bilateral weighted shifts.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth to mention that 3-isometries have been very popular recently (see e.g. [2,10,23]). Beyond the results already mentioned from [2], these operators were, for instance, utilized by Bermúdez et al as examples to shed some light on relations of certain ergodic and dynamical properties of m-isometric operators (see [10] for more details).…”

Section: Introductionmentioning

confidence: 99%

“…[2,10,23]). Beyond the results already mentioned from [2], these operators were, for instance, utilized by Bermúdez et al as examples to shed some light on relations of certain ergodic and dynamical properties of m-isometric operators (see [10] for more details). That is why we provide an explicit description of 3-isometric composition operators on directed graphs with one circuit that involves real polynomials in one variable (see Theorem 3.5).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

$$\varvec{m}$$–Isometric Composition Operators on Directed Graphs with One Circuit

Jabłoński

Kośmider

2021

Integr. Equ. Oper. Theory

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The aim of this paper is to investigate m–isometric composition operators on directed graphs with one circuit. We establish a characterization of m–isometries and prove that complete hyperexpansivity coincides with 2–isometricity within this class. We discuss the m–isometric completion problem for unilateral weighted shifts and for composition operators on directed graphs with one circuit. The paper is concluded with an affirmative solution of the Cauchy dual subnormality problem in the subclass with circuit containing one element.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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$$\varvec{m}$$–Isometric Composition Operators on Directed Graphs with One Circuit

Jabłoński

Kośmider

2021

Integr. Equ. Oper. Theory

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“…However, when infinite-dimensional space taken into account, linear operators may have dense orbits. One of the focal branches of dynamical system is ergodic theory [1], which relates analysis with probability theory and deals with measurable dynamics. It exerts measure theory to the study of the behavior of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Central Limit Theorem in View of Subspace Convex-Cyclic Operators

Hasan¹,

Ahmed²,

Hama³

et al. 2021

Bul. Kar. Univ. - Math.

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In our work we have defined an operator called subspace convex-cyclic operator. The property of this newly defined operator relates eigenvalues which have eigenvectors of modulus one with kernels of the operator. We have also illustrated the effect of the subspace convex-cyclic operator when we let it function in linear dynamics and joining it with functional analysis. The work is done on infinite dimensional spaces which may make linear operators have dense orbits. Its property of measure preserving puts together probability space with measurable dynamics and widens the subject to ergodic theory. We have also applied Birkhoff’s Ergodic Theorem to give a modified version of subspace convex-cyclic operator. To work on a separable infinite Hilbert space, it is important to have Gaussian invariant measure from which we use eigenvectors of modulus one to get what we need to have. One of the important results that we have got from this paper is the study of Central Limit Theorem. We have shown that providing Gaussian measure, Central Limit Theorem holds under the certain conditions that are given to the defined operator. In general our work is theoretically new and is combining three basic concepts dynamical system, operator theory and ergodic theory under the measure and statistics theory.

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“…Ergodic dynamical systems seem to be of interest for a few decades, with an increasing number of papers appearing lately (see, for example, [1,2,5,9,14]), a large number of them concerning convex-cyclic operators.…”

Section: Introductionmentioning

confidence: 99%

On Subspace Convex-Cyclic Operators

Woźniak¹,

Ahmed²,

Hama³

et al. 2020

Z. mat. fiz. anal. geom.

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Let H be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator T and study its important relation with the invariant subspace problem on H: the operator T is said to be subspace convex-cyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace M in a relatively dense set. We give the sufficient condition for a subspace convex-cyclic transitive operator T to be subspace convex-cyclic. We also give a special type of the Kitai criterion related to invariant subspaces which implies subspace convex-cyclicity. Finally we show a counterexample of a subspace convexcyclic operator which is not subspace convex-cyclic transitive.

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Ergodic and dynamical properties of m-isometries

Cited by 7 publications

References 22 publications

$$\varvec{m}$$–Isometric Composition Operators on Directed Graphs with One Circuit

$$\varvec{m}$$–Isometric Composition Operators on Directed Graphs with One Circuit

Central Limit Theorem in View of Subspace Convex-Cyclic Operators

On Subspace Convex-Cyclic Operators

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