### 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(MD) 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 < dimM
_{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) = λx + 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].

Original language | English |
---|---|

Journal | Linear and Multilinear Algebra |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

}

**Power commuting additive maps on rank-k linear transformations.** / Chou, Ping–Han; Liu, Cheng-Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Power commuting additive maps on rank-k linear transformations

AU - Chou, Ping–Han

AU - Liu, Cheng-Kai

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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(MD) 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 < dimM 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) = λx + 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].

AB - 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(MD) 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 < dimM 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) = λx + 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].

UR - http://www.scopus.com/inward/record.url?scp=85063930246&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063930246&partnerID=8YFLogxK

U2 - 10.1080/03081087.2019.1600465

DO - 10.1080/03081087.2019.1600465

M3 - Article

AN - SCOPUS:85063930246

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

ER -