Co-EP Banach algebra elements

Abstract

Abstract. In this work, given a unital Banach algebra A and a ∈ A such that a has a Moore-Penrose inverse a † , it will be characterized when aa † − a † a is invertible. A particular subset of this class of objects will also be studied. In addition, perturbations of this class of elements will be studied. Finally, the Banach space operator case will be also considered.

Core inverse in Banach algebras

“…Necessary and sufficient conditions which involve the dual core inverse and ensure that a Banach space operator is EP, are presented now. (xii) T # exists and T # (T # ) n T # = (T # ) n (T # ) 2 ; (xiii) T # exists, T {1, 4} = ∅ and T (1,4) (T # ) n+1 = T # T (1,4) (T # ) n , for any T (1,4) ∈ T {1, 4}; (xiv) T # exists and T (T # ) n T # + (T # ) n T T # = 2(T # ) n ; (xv) T # exists and T n T T # + T # T T n = 2T n ; (xvi) T # exists, T {1, 4} = ∅ and T n = T (1,4) T T n , for any T (1,4) ∈ T {1, 4}; (xvii) there exist T # and an invertible operator U ∈ B(X ) such that T = T # U ; (xviii) there exist T # and U ∈ B(X ) such that T = T # U ;…”

“…If a ∈ A is a Moore-Penrose invertible element such that aa † − a † a ∈ A −1 , then a is called a co-EP element. Co-EP matrices were investigated in [2], co-EP Banach algebra elements in [4] and co-EP elements of rings in [3]. In this section, we characterize co-EP Banach algebra elements using the core inverse and the dual core inverse.…”