Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{ H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${ \langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{ H},$ induces a seminorm ${ \|\cdot\|}_A$ on $\mathcal{ H}$. Let $w_A(T)$ denote the $A$-numerical radius of an operator $T$ in the semi-Hilbertian space $\big(\mathcal{ H}, { \|\cdot\|}_A\big)$. In this paper, for any semi-Hilbertian operators $T$ and $S$, we show that $w_A(TR) = w_A(SR)$ 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.
Categories: math.FA