A-numerical radius and product of semi-Hilbertian operators

Ali Zamani

paper

A-numerical radius and product of semi-Hilbertian operators

arXiv:1912.11168

Abstract

Let $A$ be a positive bounded operator on a Hilbert space $(H, ⟨ \cdot, \cdot ⟩)$ . The semi-inner product $⟨ x, y ⟩_{A} := ⟨ A x, y ⟩$ , $x, y \in H,$ induces a seminorm $∥ \cdot ∥_{A}$ on $H$ . Let $w_{A} (T)$ denote the $A$ -numerical radius of an operator $T$ in the semi-Hilbertian space $(H, ∥ \cdot ∥_{A})$ . In this paper, for any semi-Hilbertian operators $T$ and $S$ , we show that $w_{A} (T R) = w_{A} (S R)$ for all ( $A$ -rank one) semi-Hilbertian operator $R$ if and only if $A^{1/2} T = λ A^{1/2} S$ for some complex unit $λ$ . From this result we derive a number of consequences.

to appear in Bulletin of the Iranian Mathematical Society