Extensions of Jacobson's Lemma

• Autores: Corach, Gustavo; Duggal, Bhaggy; Harte, Robin
• Año de publicación: 2013
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Jacobson'S Lemma; Operator; Resolvent
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

Jacobson’s Lemma says that if a c ∈ A thenac − 1 ∈ A−1 ⇐⇒ ca − 1 ∈ A−1 which holds separately for the left and the right invertibles of A, as well as for the non zero-divisors of A. In this note, we generalize the identity above and many of its relatives from ca − 1 to certain ba − 1: specifically we will suppose aba = aca. Three special cases are of interest: the case b = c which will give Jacobson’s lemma; the case in which aba = aca = a in which both b and c are generalized inverses of a ∈ A; and the case aba = a^2 in which c = 1. This last case goes back to Vidav; in particular, Schmoeger shows that aba=a^2 holds if there are idempotents p = p^2 q = q^2 for which a = qp and b = pq. The central results in this note are of course pure algebra: but in the neighboring realm of topological algebra they have very close relatives, and we take the opportunity to extend our purely algebraic observations to their topological analogues.
Fil: Corach, Gustavo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Fil: Duggal, Bhaggy . Visegradska. Faculty of Science and Mathematics. Department of Mathematics; Serbia
Fil: Harte, Robin. Universidad de Dublin; Irlanda