COMPACT INTERTWINING RELATIONS FOR COMPOSITION OPERATORS BETWEEN THE WEIGHTED BERGMAN SPACES AND THE WEIGHTED BLOCH SPACES

Abstract

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.