We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol. 1. Introduction Let be the open unit disk in the complex plane and the space of analytic functions on . A function is said to belong to the Bloch space if Under the norm defined by ,？？ is a conformally invariant Banach space. The functions in the Bloch space satisfy the following growth condition: An important class of M？bius invariant spaces is given by the analytic Besov spaces , with , consisting of the functions such that where denotes the normalized area measure on the unit disk. The quantity is a seminorm and the Besov norm is defined by By Lemma 1.1 of [1], for , Thus, the space is continuously embedded in the Bloch space. Moreover, by Theorem 9 in [2], the functions in satisfy the growth condition Let denote the set of all functions such that where the supremum is taken over all and . By Theorem 5.3 of [3] and the Closed Graph theorem, an analytic function on belongs to if and only if . Furthermore, The quantities in (8) are just seminorms for the space , as they do not distinguish between functions differing by a linear function. The norm yields a Banach space structure on , which is called the Zygmund space. For more information on the Zygmund space on the unit disk, see, for example, [3]. Let denote the set of all analytic self-maps of . Each induces the composition operator on defined by for and . We refer the interested reader to [4, 5] for the theory of the composition operators. Let . The multiplication operator is defined as for and . The composition product of？？ and yields a linear operator on called the weighted composition operator with symbols ？？and . In recent years, considerable interest has emerged in the study of the weighted composition operators due to the important role they play in the study of the isometries on many Banach spaces of analytic functions, such as the Hardy space (for , ) [6, 7], the weighted Bergman space [8], and the disk algebra [9]. There is a very extensive literature on the composition operators and the weighted composition operators between the Bloch space and other spaces
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