Cauchy transforms of measures and weighted shift operators on the disc algebra
type: Item
instance of: scholarly article
title: Cauchy transforms of measures and weighted shift operators on the disc algebra
zbMATH Open document ID: 0863.30045
author: R. A. Hibschweiler
author: Eric Nordgren
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-06-10T00:00:00Z
full work available at URL: CONT26-2.html
review text: Let \(H(U)\) denote the space of holomorphic functions on the unit disk \(U= \{z \in \mathbb{C}:|z|<1\}\), and denote by \(M (\Gamma)\) the Banach space of complex Borel measures \(\mu\) on the unit circle \(\Gamma=\{z\in\mathbb{C}:|z|=1\}\) with the total variation norm \(|\mu |=\int_\Gamma|d\mu|\). For \(\alpha >0\), the \(\alpha\)-Cauchy transform \({\mathcal C}_\alpha\mu\) of \(\mu\in M (\Gamma)\) is given by \[ ({\mathcal C}_\alpha,\mu) (z)=\int_\Gamma (1-\zeta z)^{-\alpha} d\mu (\zeta), \quad z \in U, \] and we let \({\mathcal F}_\alpha={\mathcal C}_\alpha (M (\Gamma))\). The latter is a subspace of \(H(U)\) and is a Banach space with the norm \(|f|_\alpha=\min \{|\mu |:\mu\in M (\Gamma)\), \(f= {\mathcal C}_\alpha\mu\}\) for \(f\in {\mathcal F}_\alpha\). Moreover, for each \(\alpha >0\), the space \({\mathcal F}_\alpha\) is isometrically isomorphic to the dual space \(A(U)^*\) of the disk algebra \(A(U)\). Denote by \({\mathcal F}_{\alpha a}\) the subspace of \({\mathcal F}_\alpha\) consisting of all \({\mathcal C}_\alpha\mu\) such that \(\mu\in M (\Gamma)\) is absolutely continuous with respect to the Lebesgue measure on \(\Gamma\). Then \({\mathcal F}_{\alpha a}\) is the norm closure of the complex polynomials in \({\mathcal F}_\alpha\). \textit{T. H. MacGregor} [Indiana Univ. Math. J. 36, 109-130 (1987; Zbl 0644.30009)] has shown that for \(0<\alpha < \beta\), \({\mathcal F}_\alpha\subset {\mathcal F}_\beta\). In the present paper, the authors establish the stronger inclusion \({\mathcal F}_\alpha\subset {\mathcal F}_{\beta a}\) with the inclusion map being a compact operator of norm one. MacGregor has also shown that \({\mathcal F}_\alpha {\mathcal F}_\beta\subset {\mathcal F}_\alpha +_\beta\) for \(\alpha, \beta >0\). Using different methods, the authors prove the more precise result, namely that \(|fg |_{\alpha + \beta} \leq|f |_\alpha |g |_\beta\) whenever \(f\in {\mathcal F}_\alpha\) and \(g\in {\mathcal F}_\beta\), \(\alpha, \beta>0\). Let \({\mathca…
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 944411
zbMATH Keywords: Cauchy transform
reviewed by: Jacob Burbea
MaRDI profile type: Publication
OpenAlex ID: W2024184332
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DOI: 10.1216/RMJM/1181072077
description: scientific article; zbMATH DE number 944411