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Power commuting additive maps on rank-k linear transformations.
- Source :
-
Linear & Multilinear Algebra . Mar2021, Vol. 69 Issue 3, p403-427. 25p. - Publication Year :
- 2021
-
Abstract
- Let D be a division ring, let M be a right vector space over D and let End (M D ) be the ring of all D -linear transformations from M into M. Suppose that R is a dense subring of End (M D ) consisting of finite rank transformations and f : R → End (M D ) is an additive map. We show that if f (x) x m (x) = x m (x) f (x) for every rank-k transformation x ∈ R , where k is a fixed integer with 1 < k < dim M D and m (x) ≥ 1 is an integer depending on x, then there exist λ ∈ Z (D) and an additive map μ : R → Z (D) I such that f (x) = λ x + μ (x) for all x ∈ R , where I denotes the identity transformation on M. This gives a natural generalization of the recent results obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815; Liu and Yang, Power commuting additive maps on invertible or singular matrices. Linear Alg Appl. 2017;530:127–149] and can be regarded as an infinite-dimensional version of the Franca theorem obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815]. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LINEAR operators
*VECTOR spaces
*IDENTITY (Psychology)
*INTEGERS
*DIVISION algebras
Subjects
Details
- Language :
- English
- ISSN :
- 03081087
- Volume :
- 69
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Linear & Multilinear Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 148981987
- Full Text :
- https://doi.org/10.1080/03081087.2019.1600465