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arXiv · 2510.14004

A solution to the Cauchy dual subnormality problem for a cyclic analytic $2$-isometry with defect operator of rank two

Mandar Khasnis·Geetanjali Phatak·Vinayak Sholapurkar
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Abstract

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a $2$-isometry is subnormal. In this article, we prove that if $\mu$ is a sum of unit point mass measures at two non-antipodal points on the unit circle, then the Cauchy dual $M_z'$ of the multiplication operator $M_z$ on the Dirichlet-type space $D(\mu)$ is not subnormal. If the points are antipodal then the subnormality of the said operator has been already established in the literature. Thus, we have a complete solution to CDSP in this case.

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Mandar Khasnis, Geetanjali Phatak, Vinayak Sholapurkar. 2025-10-15. A solution to the Cauchy dual subnormality problem for a cyclic analytic $2$-isometry with defect operator of rank two. https://arxiv.org/abs/2510.14004

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