Compact intertwining relations for composition operators between the weighted bergman spaces and the weighted bloch spaces
Compact intertwining relations for composition operators between the weighted bergman spaces and the weighted bloch spaces
复制标题
DOI：
10.4134/jkms.2014.51.1.125
复制
发表时间：
2014
影响因子：
4.5
通讯作者：
C. Tong;Ze‐hua Zhou
中科院分区：
文献类型：
--
作者：
C. Tong;Ze‐hua Zhou
摘要。研究了组合算子的紧缠结关系，其缠结算子是单位盘上从加权Bergman空间到加权Bloch空间的Volterra型算子。结果，我们通过这种关系发现了加权bergman空间和小加权Bloch空间之间的新联系。1. 如果X和Y是两个巴拿赫空间，符号B（X，Y）表示从X到Y的所有有界线性算子的集合。设K（X，Y）是B（X，Y）的所有紧元素的集合，Q（X，Y）是商集B(X，Y)/K（X，Y）。对于线性算子A∈B(X，X), B∈B(Y，Y), T∈B(X，Y)，短语“T在Q（X，Y）中缠绕A和B”（或“T紧密缠绕A和B”）意味着（1.1）TA = BT模K（X，Y）与t6 = 0。符号A∝K B (T)表示式（1.1）中的关系。事实上，如果T是X上的可逆算子，那么关系∝K是对称的。我们用H(D)表示复单位盘上所有全纯函数的类，用S(D)表示复单位盘上所有全纯自映射的集合。设α >−1,0 0。这些α，β，γ和p的设置在下列情况下有效，除非另有说明。
Abstract. We study the compact intertwining relations for compositionoperators, whose intertwining operators are Volterra type operators fromthe weighted Bergman spaces to the weighted Bloch spaces in the unitdisk. As consequences, we find a new connection between the weightedBergman spaces and little weighted Bloch spaces through this relations. 1. IntroductionIf X and Y are two Banach spaces, the symbol B(X,Y ) denotes the col-lection of all bounded linear operators from X to Y . Let K(X,Y ) be thecollection of all compact elements of B(X,Y ), and Q(X,Y) be the quotientset B(X,Y )/K(X,Y ).For linear operators A ∈ B(X,X), B ∈ B(Y,Y ) and T ∈ B(X,Y ), thephrase “T intertwines A and B in Q(X,Y)” (or “T intertwines A and B com-pactly”) means that(1.1) TA = BT mod K(X,Y ) with T 6= 0 .Notation A ∝ K B (T) represents the relation in equation (1.1). In fact, if T isan invertible operator on X, then the relation ∝ K is symmetric.We denote the class of all holomorphic functions on the complex unit disk Dby H(D), and the collection of all the holomorphic self mappings of Dby S(D).Let α > −1, 0 0. These settingsof α,β,γ and p are valid in the following context unless specifications.