Univalent functions in cyclic vectors in \(Q_p\) space
type: Item
instance of: scholarly article
title: Univalent functions in cyclic vectors in \(Q_p\) space
zbMATH Open document ID: 1297.30075
author: Zhengyuan Zhuo
author: Shanli Ye
published in: Complex Analysis and Operator Theory
publication date: 2014-06-18T00:00:00Z
review text: Let \(H(\mathbb D)\) be the set of holomorphic functions on the open unit disk \(\mathbb D\) and \(X\subset H(\mathbb D)\) be a Banach space such that the polynomials are norm (or weak\(^*\)) dense in \(X\). A cyclic vector in \(X\) is a function \(f\), the products of which with polynomials are dense in \(X\). The space \(Q_p\), \( 0\leq p<\infty\), consists of functions \(f\in H(\mathbb D)\) such that \[ \|f\|_{Q_p}=\sup_{a\in{\mathbf D}}\bigg(\int_{\mathbf D}|f'(z)|^2\bigg(1-\bigg|\frac{a-z}{1-{\overline a}z}\bigg|^2\bigg)^p \mathrm{d}m(z)\bigg)^{1/2}<\infty, \] where \(m\) is the Lebesgue area measure on \(\mathbb D\), equipped with the norm \(\|f\|_*=|f(0)|+\|f\|_{Q_p}\). Note that \(Q_0\) is the Dirichlet space, \(Q_1=\mathrm{BMOA}\) and the \(Q_p\) are Bloch spaces when \(1<p<\infty\). The space \(Q_{p,0}\subset Q_p\) consists of the functions \(f\in Q_p\) such that \[ \lim_{|a|\to 1}\bigg(\int_{\mathbf D}|f'(z)|^2\bigg(1-\bigg|\frac{a-z}{1-{\overline a}z}\bigg|^2\bigg)^p \mathrm{d}m(z)\bigg)^{1/2}=0. \] The authors show that a univalent function \(f\) in \(Q_p\) (resp. \(Q_{p,0}\)) is cyclic in \(Q_p\) (resp. \(Q_{p,0}\)) if and only if it does not vanish on \(\mathbb D\).
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 30H25
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 6305066
zbMATH Keywords: univalent function
zbMATH Keywords: outer function
zbMATH Keywords: \(H^p\) space
zbMATH Keywords: \(Q_p\) space
zbMATH Keywords: cyclic vector
reviewed by: Thomas V. Tonev
MaRDI profile type: Publication
full work available at URL: s11785-013-0338-y
OpenAlex ID: W2460365599
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DOI: 10.1007/S11785-013-0338-Y
description: scientific article; zbMATH DE number 6305066