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 ∈ H, induces a seminorm || · || A on H. Let wA(T) denote the A-numerical radius of an operator T in the semi-Hilbertian space (H, || · || A). In this paper, for any semi-Hilbertian operators T and S, we show that wA(T R) = wA(SR) for all (A-rank one) semi-Hilbertian operator R if and only if A^1/2T = λA^1/2S for some complex unit λ. From this result, we derive a number of consequences.