SOME NUMERICAL RADIUS INEQUALITIES FOR PRODUCTS OF HILBERT SPACE OPERATORS

Abstract

We prove several numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones. More precisely, we prove that, if A,B ∈ B(H ) such that A is self-adjoint with λ1= minλi∈ σ (A) (the spectrum of A) and λ2= maxλi∈ σ (A). Then
ω(AB) ≤ ∥A∥ω(B) + ( ∥A∥ − 0.5 |λ1+ λ2| ) DB,
where DB= inf_{λ∈C} ∥B − λI∥. In particular, if A > 0 and σ(A) ⊆ [k∥A∥,∥A∥], then
ω(AB) ≤ (2 − k)∥A∥ ω(B).