Improved inequalities for the numerical radius via Cartesian decomposition

Abstract

We develop various lower bounds for the numerical radius w(A) of a bounded linear operator A defined on a complex Hilbert space, which improve the existing inequality w 2 (A) ≥ 1 4 A * A + AA * . In particular, for r ≥ 1, we show that 1 2i (A − A * ), respectively, stand for the real and imaginary parts of A. The numerical range of A, denoted as W (A), is defined by W (A) = { Ax, x : x ∈ H , x = 1} . We denote by A , c(A), and w(A) the operator norm, the Crawford number, and the numerical radius of A, respectively. Recall that c(A) = inf