On the $\rho$-numerical radius of rank-one operators and parametrized Buzano-type inequalities

Abstract

Let $\mathcal{H}$ be a complex Hilbert space and $\rho>0$. We establish the explicit formula \[ \omega_\rho (a\otimes b)=\frac{1}{\rho}\|a\| \|b\|+\left|1-\frac{1}{\rho}\right||\langle a,b\rangle|,\qquad a,b\in\mathcal{H}, \] for the $\rho$-numerical radius of rank-one operators, a unified expression interpolating between the operator norm, the numerical radius, and the spectral radius, which correspond to $\rho=1,2$, and $\rho\to\infty$, respectively. As an application, we derive a parametrized family of Buzano-type inequalities for four vectors. We extract from it an explicit closed-form bound and, under a reality condition which is automatically satisfied in real Hilbert spaces, we determine the best bound in the family in closed form. Examples show that the resulting bounds can be strictly sharper than both the Cauchy--Schwarz and the Buzano inequality, and the classical Buzano inequality is recovered as a boundary case, which yields an alternative proof of it.