A-Numerical Radius and Product of Semi-Hilbertian Operators

Let A be a positive bounded operator on a Hilbert space (H, <., .>). The semi-inner product < x, y >(A) := < Ax, y >, x, y is an element of H, induces a seminorm parallel to . parallel to(A) on H. Let w(A)(T) denote the A-numerical radius of an operator T in the semi-Hilbertian space (H, parallel to . parallel to(A)). In this paper, for any semi-Hilbertian operators T and S, we show that w(A)(T R) = w(A)(SR) for all ( A-rank one) semi-Hilbertian operator R if and only if A(1/2)T = lambda A(1/2)S for some complex unit lambda. From this result, we derive a number of consequences.