Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains
type: Item
instance of: scholarly article
title: Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains
zbMATH Open document ID: 1207.32014
author: Stevo Stević
published in: Applied Mathematics and Computation
publication date: 2010-11-18T00:00:00Z
review text: The author extends his former results about the norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball [Appl. Math. Comput. 215, No. 6, 2199--2205 (2009; Zbl 1186.32004)] to bounded symmetric domains \(D\). Let \(H(D)\) denote the ring of holomorphic functions on \(D\) and \[ H^p(D)= \Bigg\{f\in H(D)\mid \|f\|_{H^p(D)}:= \sup_{0\leq r<1} \bigg(\int_{\partial D}|f(r\zeta)|^p \,d\sigma(\zeta)\bigg)^{\frac{1}{p}}<\infty\Bigg\} \] the Hardy space of \(D\) for \(p>0\). Let \(u\in H(D)\) such that the multiplication operator \(M_u: H^p(D)\rightarrow H^p(D)\), \(M_uf(z):=u(z)f(z)\), is well-defined. The author calculates the norm of \(M_u\). \(M_u\) is bounded iff \(\|u\|_\infty=\sup_{z\in D}|u(z)|<\infty\), and then \(\|M_u\|=\|u\|_\infty\). More generally, let \(D=D_1\times\dots\times D_m\) be a decomposition of \(D\) with irreducible symmetric domains \(D_j\), \(1\leq j\leq m\). Let \(\varphi_j: D_j\rightarrow D_j\) be holomorphic, \(1\leq j\leq m\), \(\varphi=(\varphi_1,\dots,\varphi_m)\). Let \(\mu:D\rightarrow\mathbb R_{>0}\) be continuous and \(H^\infty_\mu(D):=\{f\in H(D)\mid \|f\|_{H^{\infty}_\mu(D)}=\sup\mu(z)|f(z)|<\infty\,\}\). Assume that, for given \(u\) and \(\mu\), the weighted composition operator \(uC_\varphi: H^p(D)\rightarrow H^\infty_\mu(D)\), \(uC_\varphi f(z):=u(z)f(\varphi(z))\), is well-defined and bounded. Then the norm of \(uC_\varphi\) can be precisely calculated in terms of \(\mu\), \(u\), \(\varphi\) and the Szegő-kernels \(S_j\) of \(D_j\), \(1\leq j\leq m\): \[ \|uC_\varphi\|=\sup_{z\in D}\mu(z)|u(z)|\prod_{j=1}^m \big(S_j(\varphi_j(z),\varphi_j(z))\big)^{\frac{1}{p}}. \]
reviewed by: Eberhard Oeljeklaus
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 5817377
zbMATH Keywords: multiplication operator
zbMATH Keywords: weighted composition operator
zbMATH Keywords: Hardy space
zbMATH Keywords: weighted-type space
zbMATH Keywords: operator norm
zbMATH Keywords: bounded symmetric domain
MaRDI profile type: Publication
full work available at URL: j.amc.2010.08.022
OpenAlex ID: W1968799726
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DOI: 10.1016/J.AMC.2010.08.022
description: scientific article; zbMATH DE number 5817377