Additive rank-1 preservers between spaces of Hermitian matrices. (English) Zbl 1160.15003

Gao, Xiang-Yu; Zhang, Xian

J. Appl. Math. Comput. 26, No. 1-2, 183-199 (2008).

Let \(\mathbb R\) be the real number field and \(\mathbb C\) the complex number field. Let \(S_{n}(\mathbb R)\) (respectively, \(H_{n}(\mathbb C))\) denote \({\mathbb R}\)-linear space of \({n}\times{n}\) real summetric (respectively, complex Hermitian) matrices; and let \(GL_{n}(\mathbb C)\) be the general linear group of \({n}{\times}{n}\) complex matrices. Let \(V=S_{m}(\mathbb R) \) or \(H_{m}(\mathbb C)\). \(\varphi:V\rightarrow H_{n}(\mathbb C)\) is called additive if \(\varphi(A+B)=\varphi(A)+\varphi(B)\), for all \(A,B\in V\). It is called linear if it is additive and satisfies \(\varphi(aA)=a\varphi(A)\) for all scalar \(a\), and \({A}\in V\) is called a rank preserver from \(V\) to \(H_{n}(\mathbb C)\), if rank \(\varphi(X)= rank(X)\) for all \({X}\in V\), and a rank-1 preserver from \(V\) to \(H_{n}(\mathbb C)\), if \(\varphi(V^1)\subseteq H_{n}{^1}(\mathbb C)\). The authors of the paper under review describe the structure of all additive rank-1 preserver maps from \(S_{m}(\mathbb R)\) (\(H_{m}(\mathbb C)\)) to \(H_{n}(\mathbb C)\).

Reviewer: Manouchehr Misaghian (Prairie View)

MSC:

15A04	Linear transformations, semilinear transformations
15A03	Vector spaces, linear dependence, rank, lineability
15B57	Hermitian, skew-Hermitian, and related matrices

Keywords:

rank; rank-1 matrix; additive rank-1 preserver; additive rank preserver; Hermitian matrices

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