The Cauchy dual subnormality problem via de Branges-Rovnyak spaces

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 ${\mathcal{M}}_z^{\prime }$ of the multiplication operator ${\mathcal{M}}_z$ 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 ${\mathcal{M}}_z^{\prime }$ 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 $\mathcal{D}(\mu )$ associated with measures $\mu$ supported on two antipodal points of the unit circle.

few typos are corrected