Cowen-Douglas function and its application on chaos

TL;DR

This paper introduces Cowen-Douglas functions on the unit disk, provides conditions for their characterization, and explores their applications in chaos theory, particularly in inverse chaos problems involving backward shift operators.

Contribution

It defines Cowen-Douglas functions on Hardy spaces and applies them to analyze chaos and inverse chaos problems in operator theory.

Findings

Established a sufficient condition for Cowen-Douglas functions.

Applied Cowen-Douglas functions to inverse chaos problems.

Connected Cowen-Douglas functions with chaos in backward shift operators.

Abstract

In this paper, on $\mathbb{D}$ we define Cowen-Douglas function introduced by Cowen-Douglas operator $M_\phi^*$ on Hardy space $\mathcal{H}^2(\mathbb{D})$ and we give a sufficient condition for Cowen-Douglas function, where $\phi\in\mathcal{H}^\infty(\mathbb{D})$. Moreover, we give some applications of Cowen-Douglas function on chaos, such as application on the inverse chaos problem for $\phi(T)$, where $\phi$ is a Cowen-Douglas function and $T$ is the backward shift operator on $\mathcal{L}^2(\mathbb{N})$.