The Cauchy dual subnormality problem via de Branges-Rovnyak spaces

TL;DR

This paper investigates when the Cauchy dual of a 2-isometry is subnormal, focusing on cyclic cases and using de Branges-Rovnyak spaces, providing new characterizations and solutions for specific Dirichlet-type spaces.

Contribution

It characterizes the subnormality of the Cauchy dual operator on de Branges-Rovnyak spaces with rational symbols, advancing understanding of the Cauchy dual subnormality problem.

Findings

Characterization of subnormality for Cauchy dual operators with rational symbols

Solution to CDSP for Dirichlet spaces with measures supported on two points

Connection established between operator theory and function space analysis

Abstract

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a $2$-isometry is subnormal. In this paper, we address this problem for cyclic $2$-isometries. In view of some recent developments in operator theory on function spaces (see \cite{AM, LGR}), one may recast CDSP as the problem of subnormality of the Cauchy dual $\mathscr M'_z$ of the multiplication operator $\mathscr M_z$ acting on a de Branges-Rovnyak space $\mathcal H(B),$ where $B$ is a vector-valued rational function. The main result of this paper characterizes the subnormality of $\mathscr M'_z$ on $\mathcal H(B)$ provided $B$ is a vector-valued rational function with simple poles. As an application, we provide affirmative solution to CDSP for the Dirichlet-type spaces $\mathscr D(\mu)$ associated with measures $\mu$ supported on two antipodal points of the unit circle.