On characterizations of isometries on function spaces
type: Item
instance of: scholarly article
title: On characterizations of isometries on function spaces
zbMATH Open document ID: 1151.32002
author: Yingbin Ruan
author: Song-Ying Li
published in: Science in China. Series A
publication date: 2008-09-09T00:00:00Z
review text: \nopagenumbers Let \(X\) be a complex Banach space. An isometric isomorphism on \(X\) is an onto linear operator \(T:X \rightarrow X\) such that \(| Tx| =| x| \) for all \(x \in X\). In the present article, the isometric isomorphisms of special spaces of holomorphic functions over the unit ball \(B_n\) in \(\mathbb C^n\) are characterized. The first is the little Bloch space \[ {\mathcal B}^0(B_n) := \{ f :B_n \longrightarrow \mathbb C: Rf(z)(1-| z| ^2) \rightarrow 0,\text{ as } z \to \partial B_n\} \] equipped with the Bloch norm \[ \| f\| _{{\mathcal B}^0(B_n)}:= \sup \{ | Rf(z)| (1-| z| ^2):z \in B_n\}. \] Here, \(Rf(z):= \sum_{j=1}^n z_j \frac{\partial f}{\partial z_j}(z) \). Theorem: An operator \(T: {\mathcal B}^0(B_n) \rightarrow {\mathcal B}^0(B_n)\) is an isometric isomorphism if and only if there exists a unitary matrix \(U\) and a number \(\theta \in [0, 2\pi)\), such that \[ Tf(z) = e^{i\theta} f(Uz) ,\qquad \forall f \in {\mathcal B}^0(B_n),\quad z \in B_n. \] For a bounded domain \(D\), let \(P:L^2(D) \rightarrow A^2(D)\) denote the Bergman projector. If \({\mathcal B}(D)\) denotes the Bloch space over \(D\), then \(P(L^\infty(D))={\mathcal B}(D)\). Now, let for \(f \in {\mathcal B}(D)\): \[ \| f\| _*:= \inf\{ \| g\| _{L^\infty(D)}:P(g)=f\}. \] Then the space \({\mathcal B}^0(B_n)\), equipped with the norm \(\| \cdot \| _*\), becomes a Banach space that is denoted by \({\mathcal B}^{\infty, -}(B_n)\). The isometric isomorphisms of \({\mathcal B}^{\infty, -}(B_n)\) are also characterized. Likewise, the isometric isomorphisms of the space VMOA of vanishing mean oscillation over \(B_n\) and the Besov spaces are completely determined.
reviewed by: Gregor Herbort
Mathematics Subject Classification ID: 32A17
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 5323343
zbMATH Keywords: isometry
zbMATH Keywords: Bloch space
zbMATH Keywords: Besov space
MaRDI profile type: Publication
full work available at URL: s11425-007-0145-x
OpenAlex ID: W2061606227
cites work: The Isometries of Hp
cites work: Q4090881
cites work: Isometries of Hp Spaces of Bounded Symmetric Domains
cites work: Isometries of Bergman Spaces over Bounded Runge Domains
cites work: On isometries of the Bloch space
cites work: On isometric isomorphisms of the Bloch space on the unit ball of \({\mathbb{C}}^ n\)
cites work: Q3596137
cites work: On the boundedness and compactness of operators of Hankel type
cites work: Factorization theorems for Hardy spaces in several variables
cites work: A note on Hardy spaces and functions of bounded mean oscillation on domains in \(\mathbb{C}^ n\)
cites work: Duality theorems for Hardy and Bergman spaces on convex domains of finite type in \({\mathbb{C}}^ n\)
cites work: On decomposition theorems for Hardy spaces on domains in \(\mathbb{C}^ n\) and applications
cites work: Multiple solutions of nonlinear elliptic equations for oscillation problems
cites work: Analysis on Besov spaces. II: Embedding and duality theorems
cites work: Analysis of some function spaces associated to Hankel operators
cites work: Trace Ideal Criteria for Composition Operators on Bergman Spaces
cites work: On bounded bilinear forms
cites work: Möbius invariant \(Q_p\) spaces associated with the Green's function on the unit ball of \(C^n\)
cites work: Möbius invariant spaces on the unit ball
cites work: Q3958740
cites work: Compactness of composition operators on the Bloch space in classical bounded symmetric domains.
cites work: Hankel Operators on the Bergman Space of Bounded Symmetric Domains
DOI: 10.1007/S11425-007-0145-X
description: scientific article; zbMATH DE number 5323343