Abstract
Let A be a bounded linear operator on a complex Hilbert space and ℜ(A) (ℑ(A)) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of A, we prove thatw(A)≥12‖A‖+12|‖ℜ(A)‖−‖ℑ(A)‖|andw2(A)≥14‖A⁎A+AA⁎‖+12|‖ℜ(A)‖2−‖ℑ(A)‖2|, where w(A) and ‖A‖ are the numerical radius and operator norm of A, respectively. We study the equality conditions for w(A)=12‖A⁎A+AA⁎‖ and prove that w(A)=12‖A⁎A+AA⁎‖ if and only if the numerical range of A is a circular disk with center at the origin and radius 12‖A⁎A+AA⁎‖. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.
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