Composition Operator Sequences And Weighted Composition Operators Defined On Analytic Function Spaces

Let F be a function space defined on a domain Ω, then compositionoperator is induced by composition of the fixed map taking Ω, into itself with functi ons in the F. It is known that the composition of functions is the fundemental operat ion and has been studied in all mathematics for a very long time. Howerver, the rese arch, as a part of liner operator theory, of operator induced by composition with a fixed mapping has a relatively short history. The study of composition operator can date to the pioneering work of E.Nordgen on the mid 1960's.The study of composition operators links analytic function theory and operator theory. Its main goal is to use some results and methods of classical analytic function theory to exploit some of the most basic questions,of which can be asked about linear operators and function spaces. At the same time, using operator theory as a tool determines the classical questions in function theory. From the pioneering work of E.Nordgen to now, composition operators have been intensively studied by many authors and a lot of profound results have been obtained. Composition operators, Which defined on function spaces such as ABMO, VMOA, Bloch-type and various of the weighted function spaces, have been studied by mathematicans during the past decade. Although a lot of results have been obtained,there are still a number of interesting questions about composition operators unsolved. There is much more to be learned about the collective compactness of composition operator sequences, essential norm of the composition operator, the compactness of various products of composition operators, spectra of the composition operators in many different settings, invariant subspace redunced by the composition operator, semigrop of the composition operators, weighted composition operators on different analytic...