Weighted Moore-Penrose Inverses In Rings With Involution

As one of the most important generalized inverses,the Moore-Penrose inverse is a key research object in the study of complex matrices,bounded linear operators over Banach spaces(Hilbert spaces),and elements in rings(semigroups).Many results have been achieved over the past few decades.The weighted Moore-Penrose inverse is a direct generalization of the Moore-Penrose inverse,which has better applicability and is more difficult to study.In recent years,the theory of weighted Moore-Penrose inverses has developed rapidly.The weighted Moore-Penrose inverse plays an increasingly important role in both theoretic aspects and applications.This paper concerns weighted Moore-Penrose inverses in rings with involution and the related EP properties.In the first part of Chapter 2,we study weighted {1,3}-invertible elements and weighted{1,4}-invertible elements in rings.The relevant results of {1,3}-,{1,4}-invertible elements are extended to the weighted cases.Weighted {1,3}-,{1,4}-invertible elements are characterized by using regularity,one-sided principal ideals,one-sided annihilators and direct sum decompositions,and various properties and construction methods of weighted ｛1,3}-,{1,2,3}-,{1,4}-,{1,2,4}-inverses are obtained.In the rest part of Chapter 2,we focus on weighted MoorePenrose invertible elements.By combining the results of weighted {1,3}-inverses and weighted{1,4}-inverses,we present the properties and characterizations of the weighted Moore-Penrose inverse,and its relationships with some other generalized inverses.Several results that have been obtained for the(weighted)Moore-Penrose inverses in rings are improved.In Chapter 3,we study EP properties based on weighted Moore-Penrose inverses.The concepts of left,right and(two-sided)weighted EP elements in rings are introduced.It is shown that the concept of weighted EP elements generalizes both that of EP elements in rings and that of weighted EP matrices and weighted EP element with respect to invertible positive elements in C*-algebras,two major weighted EP objects.Properties,representations and characterizations of left,right and two-sided weighted EP elements are obtained;Several relevant results for EP elements,weighted EP matrices and weighted EP elements in C*-algebras are generalized and developed.In addition,left,right and two-sided weighted EP properties are applied to investigate the inverse order law and the absorption law of several generalized inverses.