# 关于诱导映射的两个保持问题Two Preserver Problems on Induced Maps

Abstract: Let F be a field, Mn(F) be the set of alln × nmatrices over F. If a map f: Mn(F) → Mn(F) is defined by f:A=(aij)1→(fij((aij))), ∀A∈Mn(F) where {aij|i,j∈[1,2,...n]} are the set of func-tions on F, then f is called a map induced by {fij} on Mn(F). If AB = BA implies f(A)f(B) = f(B)f(A), then f is called preserving commutativity of matrices. If B2=B implies (f(B))2=f(B), then f is called preserving idempotent matrices. In this paper, we characterize induced maps preserving idempo-tence and commutativity of matrices over fields, resprectively.

[1] Li, C.K., Plevnik, L. and Semrl, P. (2012) Preservers of matrix pairs with a fixed inner product value. Operators and Matrices, 6, 433-464.
http://dx.doi.org/10.7153/oam-06-29

[2] Cao, C.G., Ge, Y.L. and Yao, H.M. (2013) Maps preserving classical adjoint of products of two matrices. Linear and Multilinear Algebra, 61, 1593-1604.
http://dx.doi.org/10.1080/03081087.2012.753592

[3] Semrl, P. (2008) Commutativity preserving maps. Linear Algebra and Its Applications, 429, 1051-1070.
http://dx.doi.org/10.1016/j.laa.2007.05.006

[4] Li, C.K., Semrl, P. and Sze, N.S. (2007) Maps preserving the nilpotency of products of operators. Linear Algebra and Its Applications, 424, 222-239.
http://dx.doi.org/10.1016/j.laa.2006.11.013

[5] Chooi, W.L. and Ng, W.S. (2010) On classical adjoint-commuting mappings betweenmatrix algebras 0. Linear Algebra and Its Applications, 432, 2589-2599.
http://dx.doi.org/10.1016/j.laa.2009.12.001

[6] Liu, S.W. and Zhang, G.D. (2006) Maps preserving rank 1 ma-trices over fields. Journal of Natural Science of Heilongjiang University, 23, 138-140.

[7] Yang, L., Ben, X.Z., Zhang, M. and Cao, C.G. (2014) Induced maps on matrices over fields. Abstract and Applied Analysis, 2014, Article ID: 596796, 5 p.

Top