Local Spectral Functions of Self-adjoint Operators

Abstract

We study some extensions of Loewner's theory of monotone operator functions. We diagonalize operators with reflection symmetry, that is we provide an operator theoretic model of a given structure of pure shift of infinite multiplicity and we describe the spectrum of the self-adjoint operator interms of structural properties of an operator in a Hilbert space. We discussed the Krein-like formula for singular perturbations of self-adjoint operators with applications for elliptic pseudo-differential operators. We characterized the linear maps, operators and linear transformations on symmetric matrices that preserve commulativity. We establish a local spectral theory for operators on a Banach space with spectral theory of commuting self-adjoint partial differential operators. We also consider a trace formula for self-adjoint operators associated to canonical differential expressions. We investigate some conditions that two self-adjoint operators to commute or to satisfy the Weyl function. The structure of the anticommutating self-adjoint operators restricted to injectivity is studied. We show that the compact self-adjoint operators commuting when satisfying the limit of the product of the parameterized exponentials of such operators. We consider the self-adjoint block operator matrices with non-separated diagonal entries and investigate correspondingly their Schur complements. We introduce and study the self-adjoint analytic operator functions and their local spectral function.