Remark on common properties of the products $ac$ and $ba$

Abstract

In this paper, we discuss the common properties for the products ac and ba in various categories under the condition a(ba) 2 = abaca = acaba = (ac) 2 a. We prove that generalized Jacobson's lemma and Cline's formula are suitable for generalized n-strong Drazin invertible in rings, and generalized Jacobson's lemma is suitable for left and right Fredholm operator on Banach spaces.

Generalized Cline's formula and Jacobson's lemma in a ring

“…Many authors investigated the common spectral properties for bounded linear operators over Banach spaces (see [12,15,19,18]). For bounded linear operators A, B, C and D, we note that I − AC (resp.…”