The S-spectrum of combinations of idempotents on two-sided quaternionic Banach algebra - Moulaharabbi, Somayya; Barraa, Mohamed (Ukrains’kyi Matematychnyi Zhurnal, 2025)
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 A and α, βR. We also study the invertibility of linear combinations of idempotents on A. To do this, we define the complexification A_c of A, which is regarded as a real Banach algebra, and establish the relationship between the S-spectrum of a A and the spectrum of a as an element of the complexification A_c.