The g-Drazin inverse of the sum in Banach algebras

Abstract

We explore the generalized Drazin inverse in a Banach algebra. Let A be a Banach algebra, and let a, b ∈ A d . If ab = λa π bab π then a + b ∈ A d . The explicit representation of (a + b) d is also presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., 2015, 156934.8] are extended.2010 Mathematics Subject Classification. 15A09, 47L10, 32A65.

New Kind of Banach Algebra Via Proximit structure

“…Given two normed proximit vector spaces 𝑋 and 𝑌 over a single scalar field 𝐹, we describe 𝐵𝐿(𝑋, 𝑌) as the vector proximit subspace of 𝐿(𝑋, 𝑌), which contains all bounded and continuous linear mappings from 𝑋 to 𝑌. 𝐵𝐿(𝑋, 𝑌) is commonly considered to be a normed proximit vector space with a proximit norm defined by ‖𝑇‖ 𝑝 =𝑠𝑢𝑝 𝑠𝑢𝑝 {‖𝑇(𝑉)‖: 𝑉 ∈ 𝑋 &‖𝑉‖ ≤ 1} Proposition 3. 5 Assume that 𝐵 is a normed proximit algebra. A dense sub proximit algebra of a Banach proximit algebra 𝐴, then has proximit isometric isomorphism of 𝐵 onto it.…”

“…[11] algebras of operators on Banach spaces and homomorphisms thereof provided some properties of linear systems defined over a commutative Banach algebra. The sum of two group invertible elements in a Banach algebra has a group inverse when certain new necessary and sufficient requirements [5] are met.…”