Ming-Huat Lim

Linear preservers on matrices

Abstract Let U denote either the vector space of n×n matrices or the vector space of n×n symmetric matrices over an infinite field F. In this paper we characterize linear mappings L on U that satisfy one of the following properties: (i) L(adjA)=adjL(A) for all A in U; (ii) L preserves idempotent matrices, and L(In)=In, where F is the real field R or the complex field C ; (iii) L(eA)=eL(A) for all A in U, where F= R or C .

Linear transformations on tensor spaces

Let U⊗V denote the tensor product of two finite dimensional vector spaces U and V over an infinite field. Let k be a positive integer such that kdim U and k dim V Let Dk denote the set of all non-zero elements of U⊗V of rank less than k. In this paper we study linear transformations T on U⊗V such that (TDk )⊆Dk .

ADDITIVE PRESERVERS OF TENSOR PRODUCT OF RANK ONE HERMITIAN MATRICES

Let K be a field of characteristic not two or three with an involution and F be its fixed field. Let Hm be the F-vector space of all m-square Hermitian matrices over K. Let �m denote the set of all rank-one matrices in Hm. In the tensor product space N k=1 Hmi, let N k i=1 �mi denote the set of all decomposable elements N k i=1 Ai such that Ai 2 �mi , i = 1,...,k. In this paper, additive maps T from Hm Hn to Hs Ht such that T(�mn) � (�st) ( {0} are characterized. From this, a characterization of linear maps is found between tensor products of two real vector spaces of complex Hermitian matrices that send separable pure states to separable pure states. Also classified in this paper are almost surjective additive maps L from N k=1 Hmi to Nl i=1 Hni such that L k i=1�mi � � Nl i=1 �ni where 2 � kl. When K is algebraically closed and K = F, it is shown that every linear map on Nk=1 Hmi that preserves Nk=1 �mi is induced by k bijective linear rank-one preservers on Hmi , i = 1,...,k.

A note on linear preservers of certain ranks of tensor products of matrices

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements.

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A m such that (x 1, …, x m ) ≡ (y 1, …, y m ) if there exists a permutation σ on {1, …, m} such that y σ(i) = x i for all i. Let A (m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A (m) are said to be adjacent if (x 1, …, x m−1, a) ∈ X and (x 1, …, x m−1, b) ∈ Y for some x 1, …, x m−1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A (m) to B (n) that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.

Semimodules and preservers of traceless symmetric Boolean matrices of factor rank two

Let denote the two element Boolean algebra and be the semimodule of all n-square symmetric Boolean matrices with zero diagonal. We characterize (i) subsemimodules of whose nonzero members all have factor rank 2, (ii) linear mappings from to that send distinct elements of term rank 2 to distinct elements of factor rank 2 and (iii) linear mappings from to that preserve elements of factor rank 2 and also elements of factor rank k for some