New results on common properties of the products AC and BA, II

Qingping Zeng, Kai Yan and Shifang Zhang

Mathematische Nachrichten, 2020, vol. 293, issue 8, 1629-1635

Abstract: In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If a,b,c are elements in a unital associative ring R satisfying aba=aca, then von Neumann regularity (resp. generalized Fredholmness relative to an ideal I of R) of 1−ac is converted into that of 1−ba. (ii) If A,B,C are bounded linear operators satisfying ABA=ACA, then I−AC and I−BA share common complementability of kernels and ranges. (iii) If a,b,c are elements in a unital semisimple Banach algebra A satisfying aba=aca and I is a trace ideal of A such that soc(A)⊆I⊆ kh(soc(A)), then 1−ac and 1−ba share common Fredholmness relative to I and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

Date: 2020
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