Composition Operators Acting between Some Weighted Möbius Invariant Spaces

In this paper we investigate conditions under which a holomorphic self-map of the unit disk induces a composition operator Cφ with closed range on the weighted Bloch space Blog. Also, we introduce a new class of functions the so called Flog(p, q, s) spaces. Necessary and sufficient conditions are given for a composition operator Cφ to be bounded and compact from Blog to Flog(p, q, s). Moreover, necessary and sufficient conditions for Cφ from the Dirichlet space D to the spaces Flog(p, q, s) to be compact are given in terms of the map φ.. 1 Sohag University, Faculty of Science, Department of Mathematics 82524 Sohag, Egypt; Taif University, Faculty of Science, Mathematics Department, Box 888 El-Hawiyah, El-Taif 5700, Saudi Arabia. E-mail address: [email protected] 2 Jazan University, Faculty of Science, Department of Mathematics, Jazan Saudi Arabia. E-mail address: [email protected] Date: Received: 8 March 2011; Revised: 15 August 2011; Accepted: 11 November 2011. ∗ Corresponding author . 2010 Mathematics Subject Classification. Primary 47B33; Secondary 46E15.