$q$-numerical radius of rank-one operators and the generalized Buzano inequality
Dušan Denčić, Hranislav Stanković, Mihailo Krstić, Ivan Damnjanović
TL;DR
The paper introduces the $q$-numerical radius $\omega_q$ and analyzes rank-one operators $a\otimes b$ on complex Hilbert spaces. It derives an explicit closed-form formula for all $q\in[0,1]$: $\omega_q(a\otimes b)=\frac{\|a\|\|b\|+q|\langle a,b\rangle|}{2}+\frac{\sqrt{1-q^2}}{2}\sqrt{\|a\|^2\|b\|^2-|\langle a,b\rangle|^2}$, generalizing the classical $q=1$ case and yielding a generalized Buzano inequality. The work also provides a corollary with an explicit $\lambda_q(a\otimes b)$ such that $\omega_q(a\otimes b)=\lambda_q(a\otimes b)\|a\|\|b\|$, analyzes the concavity of $q\mapsto \omega_q(a\otimes b)$, and extends these results to block-matrix forms and analytic functions of rank-one operators, broadening the scope of operator-inequality techniques in Hilbert spaces.
Abstract
Here, we study the $q$-numerical radius of rank-one operators on a Hilbert space $\mathcal{H}$. More precisely, for $q \in [0,1]$ and $a, b \in \mathcal{H}$, we establish the formula \[ ω_q(a \otimes b) = \frac{1}{2}\left(\|a\|\|b\| + q|\langle a, b \rangle| + \sqrt{1-q^2}\sqrt{\|a\|^2\|b\|^2 - |\langle a, b \rangle|^2}\right), \] which represents a generalization of the well-known formula for the numerical radius of a rank-one operator in a Hilbert space, obtained by setting $q = 1$. As a corollary, we also derive a generalization of the classical Buzano inequality.