Reverse order law for the group inverses

Abstract

This paper is to present some equivalent conditions concerning the reverse order law (AB) # = B # A # for the group invertible operators A, B on a Hilbert space H.

Further results on the reverse order law for the group inverse in rings

“…The following result was established in [4] when the setting is the Banach algebra of all bounded linear operators on a Hilbert space. However, we will establish this result in rings by using only algebraic techniques.…”

“…However, we will establish this result in rings by using only algebraic techniques. Notice that, in particular, involution and norms will not be used (which were employed in [4]). …”

“…C.Y. Deng [4] presented some equivalent conditions concerning the reverse order law (AB) # = B # A # for group invertible operators A, B on a Hilbert space. N.Č.…”

Some results on the reverse order law in rings with involution

“…Since this formula cannot be trivially extended to various generalized inverses of the product ab, the reverse order law for various generalized inverses yields a class of interesting problems that are fundamental in the theory of generalized inverses. Many authors studied these problems [1,2,[4][5][6][7].…”

“…Deng [4] presented some necessary and sufficient conditions concerning the reverse order law (ab) # = b # a # for the group invertible linear bounded operators a and b on Hilbert spaces. He used the matrix form of operators induced by some natural decomposition of Hilbert spaces.…”