A New Characterization of Generalized Weighted Composition Operators from the Bloch Space into the Zygmund Space

is a Banach space. We call Z the Zygmund space. For more information on the Zygmund space, see, for example, [1, 2]. Let S(D) denote the set of analytic self-map of D. Associated with φ ∈ S(D) is the composition operator C φ , which is defined by C φ (f)(z) = f(φ(z)) for z ∈ D and f ∈ H(D). We refer the readers to the book [3] for the theory of the composition operator on various function spaces. Let u ∈ H(D). The weighted composition operator, denoted by uC φ , is defined by

Theorem A. Let  ∈ (D), , an analytic self-map of D and  a positive integer.Then the following assertions hold.
(a) The operator   , : B(or B 0 ) → Z is bounded if and only if Here we give a new criterion for the boundedness or compactness of the operator   , ; namely, we use three families of functions to characterize the operator   , : B → Z.
Throughout the paper,  denotes a positive constant which may differ from one occurrence to the other.The notation  ≍  means that there exists a positive constant  such that / ≤  ≤ .

Main Results
In this section we give our main results and proofs.For  ∈ D, set Next, we will use these three families of functions to characterize generalized weighted composition operators   , : B → Z.
as desired.
(c)⇒(a).Suppose that  ∈ Z, and  2 and  3 and  are finite.To prove this implication, we only need to show that these conditions imply (6).A calculation shows that For the simplicity, we denote 2  ()  () + ()  () by V().
The proof of this theorem is finished.
To get the characterization of the compactness of   , : B(or B 0 ) → Z, we need the following criterion, which follows from standard arguments similar to those outlined in Proposition 3.11 of [3] (if such a sequence does not exist, then the limits in (c) automatically hold).Since the sequences { (  ), },  = 1, 2, 3, are bounded in B 0 and converge to 0 uniformly on compact subsets of D, by Lemma 2, we get ‖  ,  (  ), ‖ Z → 0, as  → ∞, which means that (c) holds.
(c)⇒(a).Suppose that the limits in (c) are 0. To prove this implication, we only need to show that (7)

Theorem 1 .
Let  ∈ (D),  be an analytic self-map of D, and  a positive integer.Then the following conditions are equivalent: B 0 → Z is bounded.Taking the functions   ,  +1 , and  +2 and using the boundedness of   , , we see that  ∈ Z, and  2 and  3 are finite.For each  ∈ D, it is easy to check that  , ∈ B 0 ,  = 1, 2, 3. Moreover ‖ , ‖ B ,  = 1, 2, 3, are bounded by constants independent of .By the boundedness of ∈D      (),    B ≤         ,

.
Lemma 2. Let  ∈ (D), , an analytic self-map of D and  be a positive integer.The operator   , : B( B 0 ) → Z is compact if and only if   , : B( B 0 ) → Z is bounded, and for any bounded sequence (  ) ∈N in B( B 0 ) which converges to zero uniformly on compact subsets of D, one has ‖  ,   ‖ Z → 0 as  → ∞.
The desired result follows.The proof of this theorem is complete.