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A solution to the Cauchy dual subnormality problem for a cyclic analytic $2-$isometry with defect operator of rank two

Mandar Khasnis, Geetanjali Phatak, Vinayak Sholapurkar

TL;DR

The paper solves the Cauchy dual subnormality problem for cyclic, analytic 2-isometries with rank-two defect by modeling them as $M_z$ on a Dirichlet-type space $D(\mu)$ with a two-point measure $\mu=\delta_{\zeta_1}+\delta_{\zeta_2}$. Using Costara's reproducing-kernel framework, it identifies $D(\mu)$ with a de Branges–Rovnyak space $H(B)$ and applies a subnormality-positivity criterion to conclude that $M_z'$ is not subnormal when the points are non-antipodal, while antipodal pairs recover known subnormality, yielding a complete finite-rank CDSP solution. The analysis hinges on explicit kernel computations, $H(B)$ realizations, and case-by-case positivity checks (including a critical cos$\theta$ case). This work clarifies the CDSP in the two-point finite-measure setting and points toward extensions to more general finitely supported measures, with potential implications for understanding Cauchy duals in de Branges–Rovnyak and Dirichlet-type frameworks.

Abstract

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a $2-$isometry is subnormal. In this article, we prove that if $μ$ is a linear combination of unit point mass measures at two non-antipodal points on the unit circle, then the Cauchy dual $M_z'$ of the multiplication operator $M_z$ on the Dirichlet space $D(μ)$ is not subnormal. If the two points are antipodal then the subnormality of the said operator has been already established in the literature. Thus, we have a complete solution of CDSP in this case.

A solution to the Cauchy dual subnormality problem for a cyclic analytic $2-$isometry with defect operator of rank two

TL;DR

The paper solves the Cauchy dual subnormality problem for cyclic, analytic 2-isometries with rank-two defect by modeling them as on a Dirichlet-type space with a two-point measure . Using Costara's reproducing-kernel framework, it identifies with a de Branges–Rovnyak space and applies a subnormality-positivity criterion to conclude that is not subnormal when the points are non-antipodal, while antipodal pairs recover known subnormality, yielding a complete finite-rank CDSP solution. The analysis hinges on explicit kernel computations, realizations, and case-by-case positivity checks (including a critical cos case). This work clarifies the CDSP in the two-point finite-measure setting and points toward extensions to more general finitely supported measures, with potential implications for understanding Cauchy duals in de Branges–Rovnyak and Dirichlet-type frameworks.

Abstract

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a isometry is subnormal. In this article, we prove that if is a linear combination of unit point mass measures at two non-antipodal points on the unit circle, then the Cauchy dual of the multiplication operator on the Dirichlet space is not subnormal. If the two points are antipodal then the subnormality of the said operator has been already established in the literature. Thus, we have a complete solution of CDSP in this case.
Paper Structure (5 sections, 1 theorem, 62 equations)

This paper contains 5 sections, 1 theorem, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reproducing kernel of $D(\mu)$
  4. Proof of the Main Theorem
  5. Epilogue

Key Result

Corollary 3.3

cgr2022 Assume the hypothesis of Theorem thm3. If $\alpha_r\overline{\alpha_t}\notin [1,\infty)$ for every $1\leq r\neq t\leq k$, then $M_z'$ is subnormal if and only if

Theorems & Definitions (6)

  • Corollary 3.3
  • proof
  • proof
  • proof
  • proof
  • proof