REFINED INEQUALITIES FOR THE NUMERICAL RADIUS OF HILBERT SPACE OPERATORS

We present some new upper and lower bounds for the numerical radius of bounded linear operators on a complex Hilbert space and show that the bounds are stronger than the existing ones. In particular, we prove that if A is a bounded linear operator on a complex Hilbert space ℋ and if ℜ(A), ℑ(A) are the real part, the imaginary part of A, respectively, then w(A)≥12∥A∥+142|∥ℜ(A)+ℑ(A)∥−∥ℜ(A)−ℑ(A)∥| and w2(A)≥14∥A*A+AA*∥+14|∥R(A)+I(A)∥2−∥R(A)−I(A)∥2|, where w(⋅) and ∥⋅∥ denote the numerical radius and the operator norm, respectively. Further, we obtain refinements of the inequalities for the numerical radius of the product of two operators. Finally, as an application of the second inequality mentioned above, we obtain an improvement of upper bound for the numerical radius of the commutators of operators.