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Left-star and right-star partial orderings. (English) Zbl 0717.15004
Für komplexe \(m\times n\)-Matrizen A, B ist die Sternordnung \(A\leq^{*}B\) durch die beiden Bedingungen \(A^*A=A^*B\) und \(AA^*=BA^*\) definiert. Verff. erklären die linke Sternordnung durch die erste der beiden Gleichungen (die alleine keine transitive Relation liefert) und die Zusatzbedingung \({\mathcal M}(A)\subset {\mathcal M}(B)\), wobei \({\mathcal M}\) den Spaltenraum der jeweiligen Matrix bedeutet. Entsprechendes gilt für die rechte Sternordnung.
Diese beiden Ordnungen werden mit anderen bekannen Ordnungsrelationen verglichen, und es werden zahlreiche gleichwertige Kennzeichnungen angegeben, die einerseits verallgemeinerte Inversen heranziehen und andererseits auch Einschränkungsmöglichkeiten für die beteiligten Matrizen untersuchen.
Die Arbeit schließt mit der Behandlung der Frage, in welchem Umfang auch andere Ordnungsrelationen zu einer linken und einer rechten Ordnung abgeschwächt werden können.
Reviewer: H.-J.Kowalsky

MSC:
15A09 Theory of matrix inversion and generalized inverses
15A30 Algebraic systems of matrices
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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