LINEAR PRESERVERS ON SPACES OF HERMITIAN OR REAL SYMMETRICAL MATRICES

Let H(n) denote the space of all n X n hermitian matrices, and L(n) the space of all n X n real symmetric matrices. Let k be a fixed positive integer, and let T be a linear map on H(n). In this paper we characterize T in each of the following cases: (1) T is rank-k nonincreasing, and Im T contains a matrix A whose rank is greater than k. (2) T preserves the inertia class (k, 0, n - k), and rank T > k2. (3) T is a rank-k preserver in case n > 2 and k = 2, or k is an odd integer. In (1) and (2) we assume k < n. In (2) we may replace H(n) by L(n) provided that the lower bound for rank T is replaced by 1/2k(k + 1). In (3) we may also replace H(n) by L(n), thus obtaining a generalization of a result of Lim, who proved the special cases k = 2 and k = n = odd number.