Centralizing additive maps on rank rblock triangular matrices

Let F be a field and let k, n1, …, nkbe positive integers with n1+ … + nk= n= 2. We denote by Tn1,…,nka block triangular matrix algebra over F with unity Inand center Z(Tn1,…,nk). Fixing an integer 1 < r= nwith r? nwhen F = 2, we prove that an additive map ?: Tn1,…nk? Tn1,…,nksatisfies ?(A)A-A?(A) ? Z(Tn1,…,nk) for every rank rmatrices A? Tn1,…,nkif and only if there exist an additive map µ: Tn1,…,nk? F and scalars ?, a? F, in which a? 0 only if r= n, n1= nk= 1 and F = 3, such that for all A= (aij) ? Tn1,…,nk, where Eij? Tn1,…,nkis 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.