The S-spectrum of combinations of idempotents on two-sided quaternionic Banach algebra

Abstract

UDC 517.5

We establish the relationship between the S-spectrum of $fg$ and the S-spectrum of $\alpha f+\beta g,$ where $f$ and $g$ are two idempotents on a two-sided quaternionic Banach algebra $\mathcal{A}$ and $\alpha, \beta\in\mathbb{R}.$ We  also study the invertibility of linear combinations of idempotents on $\mathcal{A}.$ To do this, we define the complexification $\mathcal{A}_c$ of $\mathcal{A},$ which is regarded as a real Banach algebra, and establish the relationship between the S-spectrum of $a \in \mathcal{A}$ and the spectrum of $a$ as an element of the complexification $\mathcal{A}_c.$