Cowen-Douglas function and its application on chaos

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 \(\mathcal{L}^2(\mathbb{N})\).