Imaginary Powers and General Variational Inequalities of Effective Construction of Positive Operators

Abstract

We show Stečkin’s theorem, transference, spectral decompositions, operators and examples with bounded and unbounded imaginary powers. We discuss the powers, spectrum of class operators, and the operator equations with applications .We establish the maps preserving the harmonic mean or the parallel sum of positive operators preserving Lebesgue decompositions. We study the strong convergence theorems and viscosity approximation methods for nonexpansive mappings and monotone mappings, with a general iterative method with strongly positive operators for general variational inequalities .We characterize the Krein’s differential system and generalization with the effective construction of a class of positive operators, in Hilbert space which do not admit triangular factorization.