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arXiv · 2503.05036

$q$-numerical radius of rank-one operators and the generalized Buzano inequality

Dušan Denčić·Hranislav Stanković·Mihailo Krstić·Ivan Damnjanović
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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 \[ \omega_q(a \otimes b) = \frac{1}{2}\left(\|a\|\|b\| + q|\langle a, b \rangle| + \sqrt{1-q^2}\sqrt{\|a\|^2\|b\|^2 - |\langle a, b \rangle|^2}\right), \] 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.

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Dušan Denčić, Hranislav Stanković, Mihailo Krstić, Ivan Damnjanović. 2025-03-06. $q$-numerical radius of rank-one operators and the generalized Buzano inequality. https://arxiv.org/abs/2503.05036

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