Bellman inequality for Hilbert space operators (Q389633)
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scientific article; zbMATH DE number 6248177
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Bellman inequality for Hilbert space operators |
scientific article; zbMATH DE number 6248177 |
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Bellman inequality for Hilbert space operators (English)
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21 January 2014
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Bellman inequality
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operator arithmetic mean
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operator concave
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operator decreasing
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positive linear functional
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Let \(H\) be a complex Hilbert space and let \(B(H)\) be the algebra of all bounded linear operators on \(H\). Let \(\Phi :B(H)\to B(H)\) be a positive unital linear map. The authors prove the following operator version of Bellman's inequality: Theorem. For any \(p>0\) , any \(\alpha ,\beta \geq 0\) with \(\alpha +\beta =1\) and any two self-adjoint contractions \(A,B\) in \(B(H)\), one has NEWLINE\[NEWLINE\left(\Phi (I-\alpha A-\beta B)\right)^{1/p} \geq \Phi (\alpha (I-A)^{1/p} +\beta (I-B)^{1/p} ).NEWLINE\]
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