Rudin orthogonality problem on the Bergman space
type: Item
instance of: scholarly article
title: Rudin orthogonality problem on the Bergman space
zbMATH Open document ID: 1228.46022
author: Kunyu Guo
author: Dechao Zheng
published in: Journal of Functional Analysis
publication date: 2011-05-20T00:00:00Z
review text: Let \(dA\) denote the area measure in the complex plane \(\mathbb{C}\), normalized so that the measure of the open unit disk equals 1. The Bergman space \(L^2_a(D)\) is the Hilbert space of holomorphic functions on \(D\) that are also square integrable functions on \(D\). The authors prove the following theorem: Let \(\phi\) be analytic on a neighborhood of the closed unit disk. If \(\{\phi^k: k= 0,1,\dots\}\) forms an orthogonal set in \(L^2_a(D)\), then \(\phi(z)= cz^n\) for an integer \(n\) and some constant \(c\). The authors conjecture that the above statement is true when \(\phi\) is bounded and analytic in \(D\). They also obtain as a consequence of their result that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity \(n\) if and only if its symbol is a constant multiple of the \(n\)th power of a Möbius transform. This latter result was obtained previously by the theory of the Hardy space of the bi-disk.
reviewed by: Alan Gluchoff
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 30H20
zbMATH DE Number: 5898826
zbMATH Keywords: Rudin's conjecture
zbMATH Keywords: Bergman space
zbMATH Keywords: multiplication operators
zbMATH Keywords: counting functions
zbMATH Keywords: orthogonal functions
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.03.006
OpenAlex ID: W2164253185
cites work: Orthogonal functions in \(H^\infty\)
cites work: Rudin’s orthogonality problem and the Nevanlinna counting function
cites work: Q3922907
cites work: The Nevanlinna counting functions for Rudin’s orthogonal functions
cites work: Q4707369
cites work: Q4405392
cites work: Q4059478
cites work: Q4352384
cites work: Q3523797
cites work: Measures induced by analytic functions and a problem of Walter Rudin
DOI: 10.1016/J.JFA.2011.03.006
description: scientific article; zbMATH DE number 5898826