Interpolating Blaschke products and the left spectrum of multiplication operators on the Bergman space
type: Item
instance of: scholarly article
title: Interpolating Blaschke products and the left spectrum of multiplication operators on the Bergman space
zbMATH Open document ID: 0765.30021
author: Kin Y. Li
published in: Hokkaido Mathematical Journal
publication date: 1992-12-15T00:00:00Z
review text: The paper presents material towards an answer to the following well known problem: Can every Blaschke product be uniformly approximated by interpolating Blaschke products? In the first part, we encounter D. Marshall's and D. Sarason's result that whenever the zeros of a Blaschke product \(B\) lie in a closed convex subset \(C\) of \(\mathbb{D}\cup\{1\}\), then \((B-\lambda)/(1-\bar\lambda B)\) is an interpolating Blaschke product provided that \(\lambda\) is not a cluster point of \(B\mid_ C\) at 1 and \(\lambda\not\in\{B(a)\): \(B'(a)=0\}\). Using the result that interpolating Blaschke products are dense in the set of finite products of interpolating Blaschke products, the author then derives the fact that every Blaschke product whose zeros lie on finitely many radii can be uniformly approximated by interpolating Blaschke products. In the second part, rather trivial corollaries are derived from a theorem of McDonald and Sundberg on the connection between the left spectrum of the multiplication operator on the Bergman space \(L_ a^ 2(\mathbb{D})\) and the one point Gleason parts of the spectrum of \(H^ \infty\).
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 78113
zbMATH Keywords: multiplication operator
zbMATH Keywords: interpolating Blaschke products
zbMATH Keywords: Bergman space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381413684
OpenAlex ID: W2012026550
DOI: 10.14492/HOKMJ/1381413684
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description: scientific article; zbMATH DE number 78113