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Mixed-type reverse order laws associated to \(\{1, 3, 4\}\)-inverse. (English) Zbl 1449.47007

Summary: In this paper, we study the mixed-type reverse order laws to \(\{1, 3, 4\}\)-inverses for closed range operators \(A, B\) and \(AB\). It is shown that \(B\{1, 3, 4\}A\{1, 3, 4\} \subseteq (AB)\{1, 3\}\) if and only if \(R({A^*}AB) \subseteq R (B)\). For every \(A^{(134)} \in A (1, 3, 4)\), we have \( (A^{(134)}AB)\{1, 3, 4\}A\{1, 3, 4\} = (AB)\{1, 3, 4\}\) if and only if \(R (A{A^*}AB) \subseteq R (AB)\). As an application of our results, some new characterizations of the mixed-type reverse order laws associated to the Moore-Penrose inverse and the \(\{1, 3, 4\}\)-inverse are established.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A62 Equations involving linear operators, with operator unknowns
15A09 Theory of matrix inversion and generalized inverses
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