Abstract
We characterize the boundedness and compactness of differences of the composition operators followed by differentiation between weighted Banach spaces of holomorphic functions in the unit disk. As their corollaries, some related results on the differences of composition operators acting from weighted Banach spaces to weighted Bloch type spaces are also obtained.
Highlights
Let H(D) and S(D) denote the class of holomorphic functions and analytic self-maps on the unit disk D of the complex plane of C, respectively
+ Cfu ≤ Cfu, (24). From which it follows that DCφ − DCψ : Hu∞ → HV∞ is bounded
First we suppose that DCφ − DCψ : Hu∞ → HV∞ is bounded and the conditions (25)–(27) hold
Summary
Let H(D) and S(D) denote the class of holomorphic functions and analytic self-maps on the unit disk D of the complex plane of C, respectively. There has been an increasing interest in studying the compact difference of composition operators acting on different spaces of holomorphic functions. Wolf [28, 29] characterized the boundedness and compactness of differences of composition operators between weighted Bergman spaces or weighted Bloch spaces and weighted Banach spaces of holomorphic functions in the unit disk. For each φ and ψ in S(D), we are interested in the operators DCφ − DCψ, and we characterize boundedness and compactness of the operators DCφ − DCψ between weighted Banach spaces of holomorphic functions in terms of the involved weights as well as the inducing maps. As a corollary we get a characterization of boundedness and compactness about the differences of composition operators Cφ −Cψ acting from weighted Banach spaces to weighted Bloch type spaces. The existence of this function is a consequence of Montel’s theorem as can be seen in [1]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
