A-numerical radius and product of semi-Hilbertian operators

TL;DR

This paper investigates the properties of the $A$-numerical radius in semi-Hilbertian spaces, establishing conditions under which the $A$-numerical radius remains invariant under certain operator transformations.

Contribution

It characterizes when the $A$-numerical radius of operator products remains equal, linking this to scalar multiples of operators in semi-Hilbertian spaces.

Findings

$w_A(TR) = w_A(SR)$ for all $A$-rank one $R$ iff $A^{1/2}T = ext{unit} imes A^{1/2}S$.

Provides new insights into the structure of semi-Hilbertian operators and their $A$-numerical radius.

Derives several consequences from the main characterization.

Abstract

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.