On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces

Abstract

Abstract. For two given Hilbert spaces H and K and a given bounded linear operator A ∈ L(H, K ) having closed range, it is well known that the Moore-Penrose inverse of A is a reflexive -inverse G ∈ L(K , H) of A which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator G to be the Moore-Penrose inverse of A are investigated in terms of normal, EP, bi-normal, bi-EP, -quasi-normal and r-quasi-normal and -quasi-EP and r-quasi-EP operators.

On the Moore-Penrose inverse in rings with involution

“…We recall the reader that a 2 R is called (i) bi-normal if aa a a D a aaa ; (ii) bi-EP if a 2 R and aa a a D a aaa ; (iii) l-quasi-normal if aa a D a aa; (iv) r-quasi-normal if aaa D aa a; (v) l-quasi-EP if a 2 R and aa a D a aa; (vi) r-quasi-EP if a 2 R and aaa D aa a [2,4,6]. …”

“…Malik and Thome [6] investigated weaker conditions than those given by Penrose for an operator to be the Moore-Penrose inverse of a given bounded linear operator between two Hilbert spaces, using operator matrices.…”

“…In this paper, applying a purely algebraic technique, we characterize the MoorePenrose inverse of an element in a ring with involution in terms of EP, normal, bi-EP, bi-normal, l-quasi-EP, r-quasi-EP, l-quasi-normal, r-quasi-normal and *-cancellable elements, generalizing the results from [6]. Thus, we show that the properties of operator matrices are not necessary for the characterization of the MoorePenrose inverse.…”

“…By corresponding examples in [6], it is showed that conditions (i)-(ix) of Corollary 2 are weaker that one of being self-adjoint and can be adopted to define the MoorePenrose inverse.…”