Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators

By A. Dajić, Jerry J. Koliha. 2008

51 citations recorded by the source corpus.

Abstract This paper reviews the equations ax  =  c and xb  =  d from a new perspective by studying them in the setting of associative rings with or without involution. Results for rectangular matrices and operators between different Banach and Hilbert spaces are obtained by embedding the ‘rectangles’ into rings of square matrices or rings of operators acting on the same space. Necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian, skew-hermitian, reflexive, antireflexive, positive and real-positive solutions, and the general solutions are described in terms of the original elements or operators. New results are obtained, and many results existing in the literature are recovered and corrected.

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/da141275aa6278a00c0281cf79e5e4cbe49cb774