Coherent and precoherent operators

Abstract

In this thesis we introduce and investigate new classes of operators which we call coherent and precoherent operators. These operators appear as solutions of some problems in the literature, but they also represent a generalization of some frequently studied classes of operators. After we study different properties of these new classes, we continue by considering a few interesting problems in operator theory. We consider problems about the Moore-Penrose inverse and arbitrary reflexive inverse of the sum of operators, range additivity of operators, lattice properties of the star and core partial orders on Hilbert space operators, the connection about the parallel sum of operators and their infimum in different partial orders, and one special type of operators, inspired by recently introduced disjoint range operators. Accordingly, we generalize and improve a number of results from the existing literature. One part of the thesis is dedicated to Rickart *-rings and generalizations of some presented results in the algebraic setting. We included a number of examples in order to demonstrate our statements and their possible extent: reduction of conditions, proving opposite directions, etc. In the end, we propose few problems for further research on these topics.