Products of Composition and Differentiation Operators from QK(p,q) Spaces to Bloch-Type Spaces

and Applied Analysis 3 Let D be the differentiation operator on H D , that is, Df z f ′ z . For f ∈ H D , the products of composition and differentiation operators DCφ and CφD are defined, respectively, by DCφ ( f ) ( f ◦ φ)′ f ′(φ) φ′, CφD ( f ) f ′ ( φ ) , f ∈ H D . 1.8 The boundedness and compactness of DCφ on the Hardy space were investigated by Hibschweiler and Portnoy in 11 and by Ohno in 12 . The case of the Bergman spaces was studied in 11 , while the case of the Hilbert-Bergman space was studied by Stević in 13 . In 14 , Li and Stević studied the boundedness and compactness of the operator DCφ on αBloch spaces, while in 15 they studied these operators between H∞ and α-Bloch spaces. The boundedness and compactness of the operator DCφ from mixed-norm spaces to α-Bloch spaces was studied by Li and Stević in 16 . Norm and essential norm of the operator DCφ from α-Bloch spaces to weighted-type spaces were studied by Stević in 17 . Some related operators can be also found in 18–21 . For some other papers on products of linear operators on spaces of holomorphic functions, mostly integral-type and composition operators, see, for example, the following papers by Li and Stević: 5, 22–30 . Motivated basically by papers 14, 15 , in this paper, we study the operators DCφ and CφD from QK p, q space to Bμ and Bμ,0 spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation A B means that there is a positive constant C such that B/C ≤ A ≤ CB. 2. Main Results and Proofs In this section we give our main results and proofs. For this purpose, we need some auxiliary results. The following lemma can be proved in a standard way see, e.g, in 9, Proposition 3.11 . A detailed proof, can be found, for example, in 31 . Lemma 2.1. Let φ be an analytic self-map of D. Suppose that μ is normal, p > 0, q > −2. Then DCφ or CφD : QK p, q → Bμ is compact if and only if DCφ or CφD : QK p, q → Bμ is bounded and for any bounded sequence fn n∈N in QK p, q which converges to zero uniformly on compact subsets of D, one has ‖DCφfn‖Bμ → 0 or ‖CφDfn‖Bμ → 0 as n → ∞. The following lemma can be proved similarly as 32 , one omits the details see also 2, 4 . Lemma 2.2. A closed set K in Bμ,0 is compact if and only if it is bounded and satisfies lim |z|→ 1− sup f∈K μ |z| ∣∣f ′ z ∣ 0. 2.1 Now one is in a position to state and prove the main results of this paper. 4 Abstract and Applied Analysis Theorem 2.3. Let φ be an analytic self-map of D. Suppose that μ is normal, p > 0, q > −2, and K is a nonnegative nondecreasing function on 0,∞ such that