The Cauchy dual subnormality problem via de Branges-Rovnyak spaces

2021/03/18 by Sameer Chavan, Soumitra Ghara, Chavan, Sameer +3 · 3 citations
Mathematics · #31C25 #44A60 #47B32 #47B38 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2103.10059

openalex publication_date 2021/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a 2-isometry is subnormal. In this paper, we address this problem for cyclic 2-isometries. In view of some recent developments in operator theory on function spaces (see \citeAM, LGR), one may recast CDSP as the problem of subnormality of the Cauchy dual \mathscr M'_z of the multiplication operator \mathscr M_z acting on a de Branges-Rovnyak space \mathcal H(B), where B is a vector-valued rational function. The main result of this paper characterizes the subnormality of \mathscr M'_z on \mathcal H(B) provided B is a vector-valued rational function with simple poles. As an application, we provide affirmative solution to CDSP for the Dirichlet-type spaces \mathscr D(μ) associated with measures μ supported on two antipodal points of the unit circle.

Cited by

• An example of a cyclic analytic 2-isometry with defect operator of rank 3, whose Cauchy dual is not subnormal
• A solution to the Cauchy dual subnormality problem for a cyclic analytic 2-isometry with defect operator of rank two
• Joint Complete Monotonicity of reciprocal of a polynomial in two variables

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
• An example of a cyclic analytic 2-isometry with defect operator of rank 3, whose Cauchy dual is not subnormal
• The Cauchy dual subnormality problem for cyclic 2-isometries
• A solution to the Cauchy dual subnormality problem for 2-isometries