For every 3 by 3 complex matrix A, every matrix-valued polynomial F satisfies ||F(A)|| <= 2 max_{z in W(A)} ||F(z)||, so W(A) is a complete 2-spectral set.
$q$-numerical radius of rank-one operators and the generalized Buzano inequality
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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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For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.
For every rho>0, the rho-numerical radius of a rank-one operator a⊗b equals (1/rho)||a||||b|| + |1-1/rho||⟨a,b⟩|, and this yields Buzano-type inequalities with sharper closed-form bounds.
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Square Functions and the Complete Crouzeix Conjecture in Dimension Three
For every 3 by 3 complex matrix A, every matrix-valued polynomial F satisfies ||F(A)|| <= 2 max_{z in W(A)} ||F(z)||, so W(A) is a complete 2-spectral set.
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Sharp spectral constants for scaled $q$-numerical ranges
For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.
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On the $\rho$-numerical radius of rank-one operators and parametrized Buzano-type inequalities
For every rho>0, the rho-numerical radius of a rank-one operator a⊗b equals (1/rho)||a||||b|| + |1-1/rho||⟨a,b⟩|, and this yields Buzano-type inequalities with sharper closed-form bounds.