Reverse order law for the Moore-Penrose inverse in C*-algebras

Abstract

Abstract. In this paper, several equivalent conditions related to the reverse order law for the Moore-Penrose inverse in C * -algebras are studied. Some well-known results are extended to more general settings. Then this result is applied to obtain the reverse order rule for the weighted Moore-Penrose inverse in C * -algebras.

A simultaneous canonical form of a pair of matrices and applications involving the weighted Moore–Penrose inverse

“…Related results can be found in [5,14,15]. Next theorem describes the form of both matrices A and B for which the Moore-Penrose inverse satisfies that property.…”

Centralizer’s applications to the (b, c)-inverses in rings

“…The equality (3.1) is known as the reverse order law of invertible elements. In general, the reverse order law does not hold for generalized inverses, for example, [6,7,18,25]. The following two lemmas will be useful in the sequel.…”

Mixed-type reverse order laws for generalized inverses in rings with involution

“…(6) b † (a † abb † ) † a † ∈ (ab){1, 3 Proof. The equivalences of conditions (1)-(4) follow as in [12,Theorem 2.6] for elements of C * -algebras. The rest follows from these equivalences and theorems in Section 2 and Section 3.…”

“…Many results concerning the reverse order law (ab) † = b † (a † abb † ) † a † for complex matrices appeared in Tian's papers [14] and [15], where the author used mostly properties of the rank of a complex matrices. In [12], a set of equivalent conditions for this reverse order rule for the Moore-Penrose inverse in the setting of C * -algebra is studied.…”