Additive maps on hermitian matrices

Abstract

Suppose K 6 ¼ GFð2 2 Þ is a field with proper involution and of arbitrary characteristic. Additive maps, which do not increase rank-one on hermitian matrices with entries from K are classified. The result is then used to classify additive maps that do not increase minimal rank on the set of symmetric elements, relative to general involution on a matrix algebra.

Adjacency preservers on invertible hermitian matrices II

“…By (2.1), any A ∈ Hn(Fq 2 ) of rank r is of the form A = P ( Ȧ ⊕ 0)P * for some invertible P and Ȧ ∈ HGL r (F q 2 ). Lemma 2.2 is a modification of [14,Lemma 3.1].…”

Additive rank-one nonincreasing maps on Hermitian matrices over the field GF(4)

“…Consequently, if it can be shown that rk Φ(X i ) = rk Φ(X j ) for each i, j, then the above identity implies rk Φ(X) = rk Φ(X 1 ) ≤ 1, and we could use [28 …”

Compound-Commuting Additive Maps on Matrix Spaces

“…We show, in Lemma 2.5, that every nonzero compound-commuting additive map is rank-one nonincreasing, i.e., the map sends rank one matrices to matrices of rank less than or equal to one. By using the structural results of rank-one nonincreasing additive maps on the space of block triangular matrices in [3], symmetric matrices in [8,7], and Hermitian matrices in [9,10,11], respectively, we characterize compoundcommuting additive maps on the spaces of square matrices, triangular matrices and symmetric matrices over arbitrary fields, and on the space of Hermitian matrices over a field with proper involution. We will see in Theorem 2.8 that the classification of compound-commuting additive maps on triangular matrices is quite different and essentially more complicated than the corresponding theorems on spaces of square matrices, symmetric matrices, and Hermitian matrices.…”