Cesàro-type operators on some spaces of analytic functions on the unit ball
type: Item
instance of: scholarly article
title: Cesàro-type operators on some spaces of analytic functions on the unit ball
zbMATH Open document ID: 1166.45009
author: Songxiao Li
author: Stevo Stević
published in: Applied Mathematics and Computation
publication date: 2009-03-10T00:00:00Z
review text: Consider the unit ball \(B\) in \(\mathbb{C}^n\), the space \(H(B)\) of the holomorphic functions \(f: B\to\mathbb{C}\), the radial derivative \[ {\mathcal R}f(z)= \sum^n_{j=1} z_j{\partial f\over\partial z_j}(z),\quad z= (z_1,\dots, z_n)\in B, \] the Bloch space \({\mathcal B}={\mathcal B}(B)\subset H(B)\) with the norm \[ \| f\|_{{\mathcal B}}= |f(0)|+ \sup_{z\in B}(1-|z|^2)|\nabla f(z)|<+\infty, \] the little Bloch space \({\mathcal B}_0(B)\) containing the functions \(f\in H(B)\) such that \[ \lim_{|z|\to 1} (1-|z|^2)|\nabla f(z)|= 0, \] the Banach space \[ {\mathcal Z}={\mathcal Z}(B)= \{f\in H(B);\,\sup_{z\in B} (1-|z|^2)|{\mathcal R}^2 f(z)|<+\infty\}, \] where \({\mathcal R}^2f={\mathcal R}({\mathcal R} f)\), with the norm \[ \| f\|_{{\mathcal Z}}= |f(0)|+ \sup_{z\in B}(1-|z|^2)|{\mathcal R}^2f(z)|, \] the closure \({\mathcal Z}_0\) in \({\mathcal Z}\) of the set of polynomials, and for a holomorphic map \(g: B\to \mathbb{C}\) the operators \(T_g: H(B)\to\mathbb{C}\) and \(L_g: H(B)\to \mathbb{C}\) defined by \[ T_gf(z)= \int^1_0 f(tz){\mathcal R}g(z) t^{-1}\,dt,\qquad L_gf(z)= \int^1_0{\mathcal R}f(tz)g(tz)t^{-1} \,dt \] for \(z\in B\) and \(f\in H(B)\). The authors study the boundedness and compactness of the operator \(T_g\) and \(L_g\) using \({\mathcal Z}\), \({\mathcal Z}_0\), \({\mathcal B}\) and \({\mathcal B}_0\). For example, if \(g\in H(B)\) then \(T_g:{\mathcal Z}\to{\mathcal Z}\) is bounded or compact if and only if \(g\in{\mathcal Z}\), \(L_g:{\mathcal Z}\to {\mathcal Z}\) or \(L_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is bounded if and only if \(g\in H^\infty\cap{\mathcal B}_{\log}\), \(L_g:{\mathcal Z}\to{\mathcal Z}\) or \(L_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is compact if and only if \(g= 0\), \(T_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is bounded if and only if \(T_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is compact and if and only if \(g\in{\mathcal Z}_0\). There exist results of boundedness and compactness of the operators \(T_g:{\mathcal Z}\to{\mat…
reviewed by: Dan-Mircea Borş
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 5527447
zbMATH Keywords: extended Cesàro operator
zbMATH Keywords: unit ball
zbMATH Keywords: Zygmund space
zbMATH Keywords: Bloch space
zbMATH Keywords: boundedness
zbMATH Keywords: compactness
zbMATH Keywords: spaces of analytic functions
MaRDI profile type: Publication
full work available at URL: j.amc.2008.12.006
OpenAlex ID: W1994343119
cites work: HARDY–BLOCH TYPE SPACES AND LACUNARY SERIES ON THE POLYDISK
cites work: Weighted integrals and Bloch spaces of \(n\)-harmonic functions on the polydisc
cites work: Extended Cesàro operators between different Hardy spaces
cites work: Interpolation by functions in the Bloch space
cites work: On some integral operators on the unit polydisk and the unit ball
cites work: Estimates of an integral operator on function spaces
cites work: The generalized Cesàro operator on the unit polydisk.
cites work: Addendum to the Paper “A note on Weighted bergman Spaces and the cesaro operator”
cites work: Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball
cites work: Weighted composition operators on some weighted spaces in the unit ball
cites work: Extended Cesàro operators on mixed norm spaces
cites work: Extended Cesàro operators on the Bloch space in the unit ball of \(\mathbb{C}^n\)
cites work: Extended Cesàro operators on Bergman spaces
cites work: Riemann--Stieltjes operators from \(F(p,q,s)\) spaces to \(\alpha\)-Bloch spaces on the unit ball
cites work: Integral type operators from mixed-norm spaces to α-Bloch spaces
cites work: Riemann--Stieltjes operators on Hardy spaces in the unit ball of \(\mathbb C^ n\)
cites work: Riemann–Stieltjes-type integral operators on the unit ball in
cites work: Q3531316
cites work: Generalized composition operators on Zygmund spaces and Bloch type spaces
cites work: Products of composition and integral type operators fromH∞to the Bloch space
cites work: Products of Volterra type operator and composition operator from \(H^\infty \)and Bloch spaces to Zygmund spaces
cites work: Riemann--Stieltjes operators between different weighted Bergman spaces
cites work: Q3511498
cites work: Products of integral-type operators and composition operators between Bloch-type spaces
cites work: Compact Composition Operators on the Bloch Space
cites work: Q3958740
cites work: Cesàro averaging operators
cites work: On an integral operator on the unit ball in \(\mathbb{C}^n\)
cites work: Boundedness and compactness of an integral operator on a weighted space on the polydisc
cites work: Composition operators between \(H^\infty\) and \(a\)-Bloch spaces on the polydisc
cites work: Q3544753
cites work: Q3070312
cites work: On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball
cites work: Q3083066
cites work: Extended Cesàro operators between Bloch-type spaces in the unit ball of \(\mathbf {C}^{n}\)
cites work: Spaces of Holomorphic Functions in the Unit Ball
DOI: 10.1016/J.AMC.2008.12.006
description: scientific article; zbMATH DE number 5527447