A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. (English) Zbl 1113.15302
Author’s abstract: It is shown that if \(A\) is a bounded linear operator on a complex Hilbert space, then
\[
w(A)\leq{1\over2}(\| A\|+\| A^2\|^{1/2}),
\]
where \(w(A)\) and \(\| A\|\) are the numerical radius and the usual operator norm of \(A\), respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.
Reviewer: Grosio Stanilov (Sofia)
MSC:
15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
26C10 | Real polynomials: location of zeros |
30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
47A12 | Numerical range, numerical radius |
47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |