A lower bound of the norm of the operator \(X\to AXB+BXA\) (Q2766069)
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scientific article; zbMATH DE number 1695320
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A lower bound of the norm of the operator \(X\to AXB+BXA\) |
scientific article; zbMATH DE number 1695320 |
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28 January 2004
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elementary operator
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0.8802799
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0.87130636
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0.8649663
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0.84985197
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0.8468056
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0.8417909
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0.83734685
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A lower bound of the norm of the operator \(X\to AXB+BXA\) (English)
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Let \(H\) denote a complex Hilbert space, \(L(H)\) the \(C^*\)-algebra of all bounded linear operators on \(H\), and \(M_{A,B}\) the operator on \(L(H)\) defined by NEWLINE\[NEWLINEM_{A,B}(X)= AXB,\quad\text{for every }X \in L(H).NEWLINE\]NEWLINE In this paper the authors prove that if \(A,B\in L(H)\) satisfy \(\inf_{\lambda\in \mathbb{C}}\|B-\lambda A\|= \|B\|\) or \(\inf_{\lambda\in \mathbb{C}}\|A-\lambda B\|= \|A\|\), then NEWLINE\[NEWLINE\|M_{A,B}+ M_{B,A}\|\geq \|A\|\|B\|.NEWLINE\]
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