Centralizing additive maps on rank r block triangular matrices

Document Type

Article

Publication Date

1-1-2021

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.

Keywords

centralizing map, Commuting map, Block triangular matrix, Rank, Functional identity

Publication Title

Acta Scientiarum Mathematicarum

Divisions

MathematicalSciences

Funders

FRGS Research Grant Scheme [Grant No: FRGS/1/2018/STG06/UM/02/9 (FP082-2018A)],RU Grant Scheme [Grant No: GPF027B-2018]

Volume

87

Issue

1-2

Publisher

Univ Szeged

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