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

Saee A. Joshi, Geetanjali M. Phatak, Vinayak M. Sholapurkar

TL;DR

The paper addresses the Cauchy dual subnormality problem (CDSP) by constructing a counterexample: a cyclic, analytic $2$-isometry unitarily equivalent to $M_z$ on a Dirichlet space $D()$ with $$ supported at three equi-spaced points on the unit circle, where the Cauchy dual $M_z'$ is not subnormal. The authors leverage the de Branges–Rovnyak model of $D()$ and Costara’s reproducing-kernel framework to compute the relevant kernels and inner products, and apply a subnormality criterion from prior work (cgr2022) to a cross-term sum $ extstyle extstyle igl( sum_j p_j(_r)ar{p_j(_t)}igr)$, showing it is nonzero for $r eq t$. This nonvanishing cross-term implies failure of the positivity condition needed for subnormality, yielding a counterexample to CDSP with three-point equi-spaced support. The result extends the landscape of CDSP by suggesting a broader conjecture that any three-point measure on the unit circle yields a non-subnormal Cauchy dual for $M_z$ on $D()$, guiding future investigations of Dirichlet-type spaces and their operator-theoretic properties.

Abstract

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.

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

TL;DR

The paper addresses the Cauchy dual subnormality problem (CDSP) by constructing a counterexample: a cyclic, analytic -isometry unitarily equivalent to on a Dirichlet space with supported at three equi-spaced points on the unit circle, where the Cauchy dual is not subnormal. The authors leverage the de Branges–Rovnyak model of and Costara’s reproducing-kernel framework to compute the relevant kernels and inner products, and apply a subnormality criterion from prior work (cgr2022) to a cross-term sum , showing it is nonzero for . This nonvanishing cross-term implies failure of the positivity condition needed for subnormality, yielding a counterexample to CDSP with three-point equi-spaced support. The result extends the landscape of CDSP by suggesting a broader conjecture that any three-point measure on the unit circle yields a non-subnormal Cauchy dual for on , guiding future investigations of Dirichlet-type spaces and their operator-theoretic properties.

Abstract

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

Paper Structure

This paper contains 5 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Problem
  4. Proof of the main result
  5. Epilogue

Key Result

Theorem 1

Let $\zeta_1$, $\zeta_2$ and $\zeta_3$ be three equi-spaced points on the unit circle $\mathbb{T}$. If the measure $\mu$ on the unit circle $\mathbb{T}$ is of the form $\mu=\delta_{\zeta_1}+\delta_{\zeta_2}+\delta_{\zeta_3},$ then the Cauchy dual $M_z'$ of $M_z$ on the Dirichlet space $D(\mu)$ is no

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • proof