Sharp spectral constants for scaled $q$-numerical ranges

Abstract

For $n\ge 2$, $A\in M_n(\mathbb{C})$ and $0<|q|\le 1$, let ${\Omega }_q(A)=q^{-1}{W}_q(A)$ be the scaled $q$-numerical range. We prove that for every $\gamma \ge 1$, \[ Ω_{η(γ)}(A) =\bigcup_{κ(S)\leqγ}W(S^{-1}AS), \qquad η(γ)=\frac{2}{γ+γ^{-1}}, \] where $\kappa (S)=\Vert S\Vert \Vert S^{-1}\Vert$. As a consequence, we prove the sharp inequality \[ \|p(A)\|\leq \max\!\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\} \max_{z\inΩ_q(A)}|p(z)|, \] for all polynomials $p$.