On the Moore–Penrose inverse, EP Banach space operators, and EP Banach algebra elements

Abstract

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C * -algebra elements.

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

“…Introduction. EP matrices and normal matrices, as well as EP linear operators and normal linear operators on Banach or Hilbert spaces have been investigated by many authors (see, for example, [1,2,3,4,5,7,8,9,10,11,13,15,17,18,20]). D. Mosić et al in [21,24] use the setting of rings with involution to investigate EP elements and normal elements, giving new characterizations to them and providing simpler and more transparent proofs to already existing ones.…”

EP Elements in Rings with Involution

“…Also, they showed that R † ∩ R # = R † ∩ R # . Motivated by [4,8,10,11], in this paper, we will give new equivalent characterizations such that an element in R to be an EP element.…”

Factorizations of weighted EP Banach space operators and Banach algebra elements

“…The objective of this work is to characterize weighted EP Banach space operators and weighted EP Banach algebra elements using factorizations; actually, three different kinds of factorizations will be considered (see sections [2][3][4]. Several reasons motivate this research.…”

“…Recall that according to [16,Lemma 2.1], there is at most one Moore-Penrose inverse. Concerning the Moore-Penrose inverse in Banach algebras, see [16,2,4,3]. For the original definition of the Moore-Penrose inverse for matrices, see [15].…”