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

TL;DR

This paper constructs a specific cyclic, analytic 2-isometry with a defect operator of rank 3, providing a counter-example to the Cauchy dual subnormality problem by showing its Cauchy dual is not subnormal.

Contribution

It presents the first known counter-example to the CDSP using a cyclic, analytic 2-isometry with a defect operator of rank 3.

Findings

The Cauchy dual of the multiplication operator on the Dirichlet space with three-point support is not subnormal.

Counter-example disproves the universal subnormality of Cauchy duals of 2-isometries.

Supports the conjecture that certain 2-isometries have non-subnormal Cauchy duals.

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 $D(\mu)$ is not subnormal if $\mu$ is supported at three equi-spaced points on the unit circle.