Centralizing additive maps on rank r block triangular matrices
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...
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my.um.eprints.348792022-05-10T08:26:42Z http://eprints.um.edu.my/34879/ Centralizing additive maps on rank r block triangular matrices Chooi, Wai Leong Mutalib, M. H. A. Tan, L. Y. QA Mathematics 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. Univ Szeged 2021 Article PeerReviewed Chooi, Wai Leong and Mutalib, M. H. A. and Tan, L. Y. (2021) Centralizing additive maps on rank r block triangular matrices. Acta Scientiarum Mathematicarum, 87 (1-2). pp. 63-94. ISSN 0001-6969, DOI https://doi.org/10.14232/actasm-020-586-y <https://doi.org/10.14232/actasm-020-586-y>. 10.14232/actasm-020-586-y |
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QA Mathematics Chooi, Wai Leong Mutalib, M. H. A. Tan, L. Y. Centralizing additive maps on rank r block triangular matrices |
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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. |
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Article |
author |
Chooi, Wai Leong Mutalib, M. H. A. Tan, L. Y. |
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Chooi, Wai Leong Mutalib, M. H. A. Tan, L. Y. |
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Chooi, Wai Leong |
title |
Centralizing additive maps on rank r block triangular matrices |
title_short |
Centralizing additive maps on rank r block triangular matrices |
title_full |
Centralizing additive maps on rank r block triangular matrices |
title_fullStr |
Centralizing additive maps on rank r block triangular matrices |
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Centralizing additive maps on rank r block triangular matrices |
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centralizing additive maps on rank r block triangular matrices |
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Univ Szeged |
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2021 |
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http://eprints.um.edu.my/34879/ |
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13.18916 |