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

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 $M_z^{\prime }$ of the multiplication operator $M_z$ on the Dirichlet space $D(\hat{I}¼)$ is not subnormal if $\hat{I}¼$ is supported at three equi-spaced points on the unit circle.