Saee A. Joshi | Searcharxiv

arXiv2025

An example of a cyclic analytic $2$-isometry with defect operator of rank $3$, whose Cauchy dual is not subnormal

The Cauchy dual subnormality problem (CDSP, for short) asks whether the Cauchy dual of a $2-$isometry is subnormal. In this article, we provide a counter-example to CDSP by constructing a cyclic, analytic, $2-$isometry whose defect operator is of rank $3$. In particular, we prove that the Cauchy dual $M_z'$ of the multiplication operator $M_z$ on the Dirichlet space $D(μ)$ is not subnormal if $μ$ is supported at three equi-spaced points on the unit circle.

math.FA↗

arXiv2024

Completely hyperexpansive operators with finite rank defect operator and de Branges-Rovnyak spaces

The process of identifying a Dirichlet-type space $D(μ)$ for a positive, Borel measure $μ$, supported on the unit circle $\mathbb T,$ with a de Branges-Rovnyak space was initiated by Sarason. A characterization of the symbol for a de Branges-Rovnyak spaces for which the shift operator is a $2$-isometry, was provided in an article by Kellay and Zarrabi. In this paper, capitalizing on the Aleman's model for the cyclic, analytic, completely hyperexpansive operators, we provide a characterization of cyclic, analytic, completely hyperexpansive operator with finite rank defect operator in terms of the symbol for a de Branges-Rovnyak space.

math.FA↗