New characterization of $(b,c)$-inverses through polarity

TL;DR

This paper introduces the concept of $(b,c)$-polar elements in associative rings, establishing their properties and equivalence to $(b,c)$-invertibility, and extends the theory to Banach spaces.

Contribution

It defines $(b,c)$-polar elements, provides necessary and sufficient conditions for their existence, and generalizes previous notions of polarity in algebra and analysis.

Findings

$(b,c)$-polar elements are equivalent to $(b,c)$-invertible elements.

Characterizations of $(b,c)$-polar elements are extended to Banach spaces.

The notion generalizes polarity concepts from prior research.

Abstract

In this paper we introduce the notion of $(b,c)$-polar elements in an associative ring $R$. Necessary and sufficient conditions of an element $a\in R$ to be $(b,c)$-polar are investigated. We show that an element $a\in R$ is $(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the $(b,c)$-polarity is a generalization of the polarity along an element introduced by Song, Zhu and Mosi\'c [14] if $b=c$, and the polarity introduced by Koliha and Patricio [10]. Further characterizations are obtained in the Banach space context.