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Numerical radius inequalities for Hilbert space operators

Fuad Kittaneh — 2005

Studia Mathematica

It is shown that if A is a bounded linear operator on a complex Hilbert space, then 1/4 ||A*A + AA*|| ≤ (w(A))² ≤ 1/2 ||A*A + AA*||, where w(·) and ||·|| are the numerical radius and the usual operator norm, respectively. These inequalities lead to a considerable improvement of the well known inequalities 1/2 ||A|| ≤ w(A) ≤ || A||. Numerical radius inequalities for products and commutators of operators are also obtained.

A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix

Fuad Kittaneh — 2003

Studia Mathematica

It is shown that if A is a bounded linear operator on a complex Hilbert space, then w ( A ) 1 / 2 ( | | 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.

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