Linear preservers of idempotence on triangular matrix spaces over any field

Abstract

Let F be any field and let T n (F) be the n × n upper triangular matrix space over F. We denote the set of all n × n upper triangular idempotent matrices over F by P n (F). A map ϕ on T n (F) is called a preserver of idempotence if ϕ(P n (F)) ⊂ P n (F); and a strong preserver of idempotence if ϕ(P n (F)) = P n (F). In this paper, we characterize the bijective linear preservers of idempotence on T n (F). Further, the strong linear preservers of idempotence on T n (F) are characterized. Mathematics Subject Classifications: 15A04; 15A03

Potent preservers of incidence algebras

“…It follows from [29, Corollary 4 and Theorem 5] that automorphisms and antiautomorphisms are the only bijective linear idempotent preservers of the uppertriangular matrix algebra T n (C). Xu, Cao and Tang [36] generalized this result to the case of an arbitrary field. As it was for M n (F ), in characteristic 2 shift maps appear in the description.…”