Cyclic Analytic 2-isometry of Finite Rank and Cauchy Dual Subnormality Problem

TL;DR

This paper constructs a new counterexample of a cyclic 2-isometric operator with non-subnormal Cauchy dual, challenging previous assumptions and employing advanced computational techniques.

Contribution

It introduces a novel counterexample to the Cauchy dual subnormality problem using analytic cyclic 2-isometries and de Branges-Rovnyak space realizations.

Findings

Counterexample of non-subnormal Cauchy dual for cyclic 2-isometry

Use of de Branges-Rovnyak space for operator realization

Numerical methods applied to polynomial root analysis

Abstract

The Cauchy dual subnormality problem has attracted the attention of the researchers in recent years. In this article, we describe the problem and present a new counter example to the problem by constructing a family of analytic, cyclic 2-isometric operators whose Cauchy dual is not subnormal. The said example is in contrast with the class of operators whose Cauchy dual is subnormal, found in the paper "The Cauchy dual subnormality problem via de branges-rovnyak spaces" by Chavan S.,Ghara S., Reza M. In the process, we employ the technique of realizing a 2-isometry as a shift operator on a suitable de Branges-Rovnyak space. The construction of the counter example involves some unwieldy computations around the roots of a polynomial of degree four. We therefore use numerical techniques with the help of SageMath to facilitate the computational work.