$C^{*}$ algebra and inverse chaos

TL;DR

This paper explores the chaotic behavior of invertible linear operators and their inverses or adjoints on Hilbert spaces, providing criteria for Lebesgue operators and Cowen-Douglas functions, along with examples illustrating chaos asymmetry.

Contribution

It introduces new criteria for Lebesgue operators and adjoint multipliers of Cowen-Douglas functions, and presents examples of operators where chaos is not preserved under inversion.

Findings

Criteria for Lebesgue operators on separable Hilbert spaces.

Criteria for adjoint multipliers of Cowen-Douglas functions.

Examples of invertible operators chaotic but their inverses are not.

Abstract

If an invertible linear dynamical systems is Li-York chaotic or other chaotic, what's about it's inverse dynamics? what's about it's adjoint dynamics? With this unresolved but basic problems, this paper will give a criterion for Lebesgue operator on separable Hilbert space. Also we give a criterion for the adjoint multiplier of Cowen-Douglas functions on $2$-th Hardy space. Last we give some chaos about scalars perturbation of operator and some examples of invertible bounded linear operator such that $T$ is chaotic but $T^{-1}$ is not.