# A solution to the Cauchy dual subnormality problem for 2-isometries

## TL;DR

This paper demonstrates that the Cauchy dual of a 2-isometry is not necessarily subnormal by providing counterexamples, thereby resolving a long-standing open problem in operator theory.

## Contribution

It provides the first counterexamples to the Cauchy dual subnormality problem for 2-isometries, clarifying conditions under which the dual operator is subnormal.

## Key findings

- Counterexamples show the Cauchy dual of a 2-isometry can be non-subnormal.
- The kernel condition is not sufficient for subnormality of the Cauchy dual.
- Certain classes like quasi-Brownian isometries always have subnormal Cauchy duals.

## Abstract

The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{\prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if $T$ is a $2$-isometry satisfying the kernel condition or a quasi-Brownian isometry, then $T^{\prime}$ is subnormal. We construct a $2$-isometric adjacency operator $T$ of a rooted directed tree such that $T$ does not satisfy the kernel condition, $T$ is not a quasi-Brownian isometry and $T^{\prime}$ is subnormal.