Measures and Adjoints of Composition Operators on Hilbert Spaces of Analytic Functions

Abstract

We study the composition operators and some interpolation problems for analytic functions on Hilbert spaces. We show a refinement of the von Neumann double commutant theorem. We describe the spectrum of the composition operator in the case when the inner function is a linear fractional transformation of the unit disc onto itself . We show the relations between the invariant subspaces of an operator and an algebra associated with the operator. We give an approximate point spectrum of a weighted shift operator and also show some invariant σ – algebras for measure preserving transformations .We consider the composition operators as integral operators and relating the composition operators an many different weighted Hardy spaces. We show the adjoins of a class of composition operators on the Bergman and Dirichlet spaces on the unit disc, with an essentially normal composition operators on Bergman spaces .We investigate the measurable transformation in finite measure spaces and mixing properties of Markov operators with ergodic transformations and ergodicity of the Cartesian products and unitary eigenopertors of ergodic transformations. We establish the analytic functions of class Hardy .we show and determine the Universal operators. We obtain the characterizations of the invariant subspaces. We study the intertwining analytic Toeplitz operators and their related composition operators with Algebras generated by same composition operators. We also study the invert ability of composition operators on the hardy spaces. Some results of the adjoins of linear fractional composition operator on the Dirichlet space are considered. Anew class of operators and a description of adjoin of composition operators are composition operators are discussed. We show which linear fractional composition operators are essentially normal. We determine the spectra of composition operators from linear fractional maps acting upon the Dirichlet space, and the adjoins of composition operators on Hilbert spaces of analytic functions.