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On the \(\mathop\leq\limits^\sharp\)-order on the set of linear bounded operators in Banach space. (English. Russian original) Zbl 1303.47003
Math. Notes 93, No. 5, 784-788 (2013); translation from Mat. Zametki 93, No. 5, 794-797 (2013).
Extending the relation \(\leq ^{\sharp}\) defined by S. K. Mitra [Linear Algebra Appl. 92, 17–37 (1987; Zbl 0619.15006)] from \(M_n(\mathbb C)\) to \(B(X)\), the author defines for \(A,B\in B(X)\) that \(A\leq^{\sharp} B\) if there exists an idempotent \(P\in B(X)\) such that \(\overline{\text{im}A}= \text{im}P\), \(\ker A =\ker P\), \(PA = PB\), and \(AP = BP\), and shows that \(\leq^{\sharp}\) is a partial order on \(B(X)\).

MSC:
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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