Partial isometries and EP elements in rings with involution

Abstract

Abstract. If R is a ring with involution, and a † is the Moore-Penrose inverse of a ∈ R, then the element a is called:In this paper, characterizations of partial isometries, EP elements and star-dagger elements in rings with involution are given. Thus, some well-known results are extended to more general settings.

On EP elements, normal elements and partial isometries in rings with involution

“…D. Mosić and D.S. Djordjevć also characterize partial isometries in terms of the pure theory of rings, generalizing known results for complex matrices (see [22,23]). …”

“…It should be noted that the Moore-Penrose inverse and the group inverse are useful in solving overdetermined systems of linear equations, and the importance of EP elements lies in the fact that they are characterized by the commutativity with their Moore-Penrose inverses. There are close connections among EP elements, partial isometries and normal elements in rings with involution (see [22,23]). …”

Weighted partial isometries and weighted-EP elements in C*-algebras

“…Since the left side of equalities (9) and (12) are equal, we observe that a * f,e a # a # = a # a * f,e a # and (xiv) is satisfied.…”

“…Inspired by [2], characterizations of partial isometries, EP elements and star-dagger elements in rings with involution were investigated in [12].…”

On the elements aa† and a†a in

“…There is no simple relation (except when a satisfies some concrete relation, see e.g. [7]) between a # and a † . One can guess that (a # ) † = (a † ) # .…”