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A note on norm estimates of the numerical radius. (English) Zbl 1144.15021

As claimed by the author, most of the results in this note are known. His purpose here is to give alternative and simpler proofs for them. For a bounded linear operator \(A\) on a Hilbert space \(H\), its numerical radius \(w(A)\) is by definition the quantity \(\sup\{|\langle Ax, x\rangle|: x\in H\), \(\| x\|= 1\}\). The author first gives a proof of \(\| A\|= 2w(A)\) for an operator \(A\) satisfying \(A^2= 0\). In fact, in the literature more precise information is known on the numerical range \(W(A)\) \((\equiv\{\langle Ax, x\rangle: x\in H\), \(\| x\|= 1\})\) for any quadratic operator \(A\) (\(A\) satisfying \(A^2+ aA+ bI= 0\) for some scalars \(a\) and \(b\)). This is in S.-H. Tso and P. Y. Wu’s matricial ranges of quadratic operators [Rocky Mt. J. Math. 29, No. 3, 1139–1152 (1999; Zbl 0957.47005)] and is proved along a similar line as here by using a block matrix representation of \(A\).
In the second part of the note, the author considers normaloid operators \(A\) (\(A\) satisfying \(\| A\|= r(A)\equiv\sup\{|\lambda|: \lambda\in\Sigma(A)\})\). In particular, he shows that if \(A\) acts on a two-dimensional space, then \(\| A^2\|=\| A\|^2\) if and only if \(A\) is normal.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A12 Numerical range, numerical radius
15B57 Hermitian, skew-Hermitian, and related matrices

Citations:

Zbl 0957.47005
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References:

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