Cowen-Douglas function and its application on chaos

Abstract

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