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Preservers of the left-star and right-star partial orders. (English) Zbl 1445.15019

Let \(\mathbb{M}_{m,n}(\mathbb{F})\) denote the set of all matrices of order \(m \times n\) over the field \(\mathbb{F}\), which is either \(\mathbb{R}\) or \(\mathbb{C}\). When \(m=n\), we denote \(\mathbb{M}_{m,n}(\mathbb{F})\) by \(\mathbb{M}_n(\mathbb{F})\). The star partial order is defined for \(A,B \in \mathbb{M}_n(\mathbb{F})\) and is denoted by \(A \underset{*}{\leq} B\), if \(A^*A=A^*B\) and \(AA^*=BA^*\). The left-star partial order is: \(A~ *\!\leq B\), if \(A^*A=A^*B\) and \(R(A) \subseteq R(B)\), whereas the right-star partial order is: \(A \leq \!* ~B\), if \(AA^*=BA^*\) and \(R(A^*) \subseteq R(B^*)\). Here, \(R(\cdot)\) denotes the range space. A map \(\Phi:\mathbb{M}_n(\mathbb{F}) \rightarrow \mathbb{M}_n(\mathbb{F})\) is said to preserve a partial order \(\leq\) in both directions if one has that \(A \leq B\) if and only if \(\Phi(A) \leq \Phi(B)\).
In this work, the authors characterize surjective maps on \(\mathbb{M}_n(\mathbb{F})\) that are not necessarily additive, such that \(\Phi\) preserves either the right-star order or the left-star order in both directions.

MSC:

15A86 Linear preserver problems
15A04 Linear transformations, semilinear transformations
15A09 Theory of matrix inversion and generalized inverses
06A06 Partial orders, general
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References:

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