The Cauchy dual subnormality problem via de Branges–Rovnyak spaces
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, 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 holomorphic 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.