A-numerical radius inequalities for semi-Hilbertian space operators

Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H induces a semi-norm ‖⋅‖A on H. Let ‖T‖A and wA(T) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper, we prove the following characterization of wA(T)wA(T)=supα2+β2=1⁡‖αT+T♯A2+βT−T♯A2i‖A, where T♯A is a distinguished A-adjoint operator of T. We then apply it to find upper and lower bounds for wA(T). In particular, we show that12‖T‖A≤max⁡{1−|cos⁡|A2T,22}wA(T)≤wA(T), where |cos⁡|AT denotes the A-cosine of angle of T. Some upper bounds for the A-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given.