Chaos for the Dynamics of Toeplitz Operators

Rubén Alejandro Martínez-Avendaño,Salud Bartoll,Ronald Richard Jiménez-Munguía,Alfred Peris

doi:10.3390/math10030425

Rubén Alejandro Martínez-Avendaño, Salud Bartoll + Show 2 more

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https://doi.org/10.3390/math10030425

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Abstract

Chaotic properties in the dynamics of Toeplitz operators on the Hardy–Hilbert space H2(D) are studied. Based on previous results of Shkarin and Baranov and Lishanskii, a characterization of different versions of chaos formulated in terms of the coefficients of the symbol for the tridiagonal case are obtained. In addition, easily computable sufficient conditions that depend on the coefficients are found for the chaotic behavior of certain Toeplitz operators.

Highlights

IntroductionHypercyclic (that is, topologically transitive) and chaotic operators on separable Banach spaces have been studied for more than twenty years (the reader is referred to the work in [1,2] for good sources on linear dynamics)
Hypercyclic and chaotic operators on separable Banach spaces have been studied for more than twenty years
It is known that analytic Toeplitz operators, that is, operators whose symbol is in H∞

Summary

Introduction

Hypercyclic (that is, topologically transitive) and chaotic operators on separable Banach spaces have been studied for more than twenty years (the reader is referred to the work in [1,2] for good sources on linear dynamics). With the above identification of H2(D) with 2, the anti-analytic Toeplitz operator TΦ with Φ(z) = ∑n≤0 anzn can be formally represented by TΦ = ∑ a−nBn, n≥0 where B is the backward shift B(x0, x1, ...) = (x1, x2, ...), so that an anti-analytic Toeplitz operator can be viewed as an upper triangular infinite matrix with constant diagonals With this identification, Bourdon and Shapiro [9] studied the dynamics of anti-analytic Toeplitz operators in the Bergman space, and Martínez [10] in more general sequence spaces. Baranov and Lishanskii [14], inspired by the work of Shkarin [15], studied hypercyclicity of Toeplitz operators on H2(D) with symbols of the form p(1/z) + φ(z), where p is a polynomial and φ ∈ H∞ They showed necessary conditions and sufficient conditions for hypercyclicity which almost coincide in the case the degree of p is one. A rich variety of chaotic properties, in the topological and in the measure theoretical sense are provided, which must be compared with previous works on the dynamics of Toeplitz operators, dealing with hypercyclicity and/or Devaney chaos

Preliminaries and Notation

Tridiagonal Toeplitz Operators

Toeplitz Operators with General Analytic Part

It is known that m

Conclusions and Future Work