Local Spectral Function of Self Adjoint Analytic Operator Functions and Invertability and Adjoints of Composition Operators on Functional Hilbert Spaces

Abstract

We show the essential norm of composition and weighted composition operators on the Hardy space and between ∝-Bloch space and β-Bloch space in polydiscs. We characterize the adjoint of a composition operator and show the basis properties of self-adjoint operator functions. We investigate the essential norm of composition operator between generalized Bloch spaces in polydiscus and its applications. We show the self-adjoint block operator matrices with non- separated diagonal entries and their Schur. complements. We discuss the methods of self-adjoint analytic operator functions and their local spectral function. We show the bases of reproducing Kernels in some model spaces and the Berezin symbol and inevitability of operators on the functional Hilbert spaces. We also show the extermal non-Compactness of composition operators with linear fractional symbol.The norm of a composition operator with linear symbol acting on the Dirichlet space is considered. We also show that the adjoints of linear fractional and spectra of composition operators are acting on the Dirichlet space. We give a characterization of adjoints of composition operators Hilbert space of analytic functions and on Hardy space of the half-plane.