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

2025/11/06 by Saee A. Joshi, Joshi, Saee A., Geetanjali M. Phatak +3
Mathematics · #Holomorphic and Operator Theory #Advanced Banach Space Theory #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2511.04565

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(μ) is not subnormal if μ is supported at three equi-spaced points on the unit circle.

Citations

• A solution to the Cauchy dual subnormality problem for a cyclic analytic 2-isometry with defect operator of rank two
• The Cauchy dual subnormality problem via de Branges-Rovnyak spaces
• de Branges-Rovnyak spaces: basics and theory
• De Branges–Rovnyak spaces and Dirichlet spaces
• Local Dirichlet spaces as de Branges-Rovnyak spaces

Related

• A solution to the Cauchy dual subnormality problem for a cyclic analytic 2-isometry with defect operator of rank two
• Cyclic Analytic 2-isometry of Finite Rank and Cauchy Dual Subnormality Problem
• The Cauchy dual subnormality problem for cyclic 2-isometries
• The Cauchy dual subnormality problem via de Branges-Rovnyak spaces
• A solution to the Cauchy dual subnormality problem for 2-isometries