Unitarily invariant norms related to the numerical radius on \(B(H)\) (Q2853304)
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scientific article; zbMATH DE number 6217267
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Unitarily invariant norms related to the numerical radius on \(B(H)\) |
scientific article; zbMATH DE number 6217267 |
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21 October 2013
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unitarily invarian norm
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numerical radius
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operator norm
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Unitarily invariant norms related to the numerical radius on \(B(H)\) (English)
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It is well known that for every operator \(A\) acting on a Hilbert space \(H\), NEWLINE\[NEWLINE\frac{1}{2}\|A\|\leq w(A)\leq \|A\|.\tag{*} NEWLINE\]NEWLINE Here, \(\|A\|\) denotes the operator norm of \(A\) and \(w(A)\) is its numerical radius, that is, NEWLINE\[NEWLINEw(A)=\sup\{ |(Ax,x)| : x\in H, \;\|x\|=1\}.NEWLINE\]NEWLINE The authors show that \(\|\cdot\|\) is the maximal (minimal) unitarily invariant norm satisfying the left (resp., right) inequality in (*). This generalizes from the finite-dimensional setting \textit{T. Ando}'s result in [Linear Algebra Appl. 417, No. 1, 3--7 (2006; Zbl 1107.15023)].
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