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

Sano, Takashi

Proc. Japan Acad., Ser. A 84, No. 1, 5-7 (2008).

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.

Reviewer: Pei Yuan Wu (Hsinchu)

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

Keywords:

numerical radius; normaloid matrix; norm estimates; quadratic operator

Citations:

Zbl 0957.47005

References:

[1]	R. Bouldin, The numerical range of a product II, J. Math. Anal. Appl. 33 (1971), 212-219. · Zbl 0211.44401
[2]	K. E. Gustafson and D. K. M. Rao, Numerical Range , Springer, New York, 1997. · Zbl 0362.47001
[3]	U. Haagerup and P. de la Harpe, The numerical radius of a nilpotent operator on a Hilbert space, Proc. Amer. Math. Soc. 115 (1992), no. 2, 371-379. · Zbl 0781.47014
[4]	P. R. Halmos, A Hilbert Space Problem Book , 2nd Ed., Springer, New York, 1982.
[5]	F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), no. 1, 11-17. · Zbl 1113.15302
[6]	V. Pták, Norms and spectral radius of matrices, Czechoslovak Math. J. 12 (87) (1962), 555-557. · Zbl 0116.25301