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

Abstract

AbstractIn 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 are elements in a unital associative ring satisfying , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal of ) of is converted into that of . (ii) If are bounded linear operators satisfying , then and share common complementability of kernels and ranges. (iii) If are elements in a unital semisimple Banach algebra satisfying and is a trace ideal of such that soc kh(soc, then and share common Fredholmness relative to and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.