Centralizing additive maps on rank r block triangular matrices

Abstract

Let F be a field and let k, n(1), ..., n(k) be positive integers with n(1) + ...+ n(k) = n >= 2. We denote by Tn(1), ..., n(k) a block triangular matrix algebra over F with unity I-n and center Z(Tn(1), ..., n(k)). Fixing an integer 1 < r <= n with r not equal n when vertical bar F vertical bar = 2, we prove that an additive map psi: Tn(1), ..., n(k) -> Tn(1), ..., n(k )satisfies psi(A)A - A psi(A) is an element of Z(TTn(1), ..., n(k)) for every rank r matrices A is an element of Tn(1), ..., n(k )if and only if there exist an additive map mu: Tn(1), ..., n(k) -> F and scalars lambda, alpha is an element of F, in which alpha not equal 0 only if r = n, n(1) = n(k) = 1 and vertical bar F vertical bar = 3, such that psi(A) = lambda A + mu(A)I-n + alpha(a(11) + a(nn))E-1n for all A = (a(ij)) is an element of T-n1, ..., n(k), where E-ij is an element of Tn(1), ..., n(k) is the matrix unit whose (i, j)th entry is one and zero elsewhere. Using this result, a complete structural characterization of commuting additive maps on rank s > 1 upper triangular matrices over an arbitrary field is addressed.