The S-spectrum of combinations of idempotents on two-sided quaternionic Banach algebra - Moulaharabbi, Somayya; Barraa, Mohamed; Moulaharabbi, Somayya; Barraa, Mohamed (Institute of Mathematics, NAS of Ukraine, 2026)
UDC 517.5 We establish the relationship between the S-spectrum of $fg$ and the S-spectrum of $α f+β g,$ where $f$ and $g$ are two idempotents on a two-sided quaternionic Banach algebra $\mathcal{A}$ and $α, β\in\mathbb{R}.$ 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 $\mathcal{A}_c.$