Products of Composition and Differentiation Operators from Bloch into Q K Spaces

C φ (f) = f ∘ φ for f ∈ H(Δ). The problem of characterizing the boundedness and compactness of composition operators on many Banach spaces of analytic functions has attracted lots of attention recently, for example, [2] and the reference therein. Let D be the differentiation operator on H(Δ); then we have Df(z) = f(z). For f ∈ H(Δ), the products of differentiation and composition operators DC φ and C φ D are defined by


Introduction and Motivation
Let Δ be the open unit disk in the complex plane and let (Δ) be the class of all analytic functions on Δ.Let () be the Euclidean area element on Δ.The Bloch space B on Δ is the space of all analytic functions  on Δ such that         B =      (0)     + sup ∈Δ (1 − || 2 )        ()      < ∞.
Under the above norm, B is a Banach space.Let B 0 denote the subspace of B consisting of those  ∈ B for which (1 − || 2 )  || → 0 as || → 1.This space is called the little Bloch space.
Throughout this paper, we assume that  : [0, ∞) → [0, ∞) is a nondecreasing and right-continuous function.A function  ∈ (Δ) is said to belong to   space (see [1] From [1], we know that Let  denote a nonconstant analytic self-map of Δ. Associated with  is the composition operator   defined by   () =  ∘  for  ∈ (Δ).The problem of characterizing the boundedness and compactness of composition operators on many Banach spaces of analytic functions has attracted lots of attention recently, for example, [2] and the reference therein.
Recall that a linear operator  :  →  is said to be bounded if there exists a constant  > 0 such that ‖()‖  ≤ ‖‖  for all maps  ∈ .And  →  is compact if it takes bounded sets in  to sets in  which have compact closure.For Banach spaces  and  of (Δ),  is compact from  to  if and only if for each sequence {  } in ; the sequence {  } ∈  contains a subsequence converging to some limit in .
Considering the definition of   spaces and   ⊆ B with some conditions, it is difficult to study the operator    from Bloch spaces to   spaces.In this paper, some sufficient and necessary conditions for the boundedness and compactness of this operator are given.
for  ∈ Δ; then for  ∈ Δ.Since Thus for all  ∈ Δ.This completes the proof of this theorem.

The Compactness
The following lemma can be proved similarly to [11].