Numerical radius and distance to unitaries

Catalin Badea (1) , Michel Crouzeix (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 IRMAR - Institut de Recherche Mathématique de Rennes

Denote by $w(A)$ the numerical radius of a bounded linear operator $A$ acting on Hilbert space. Suppose that $A$ is invertible and that $w(A)\leq 1{+}\varepsilon$ and $w(A^{-1})\leq 1{+}\varepsilon$ for some $\varepsilon\geq0$. It is shown that $\inf\{\|A{-}U\|\,: U$ unitary$\}\leq c\varepsilon^{1/4}$ for some constant $c>0$. This generalizes a result due to J.G.~Stampfli, which is obtained for $\varepsilon = 0$. An example is given showing that the exponent $1/4$ is optimal. The more general case of the operator $\rho$-radius $w_{\rho}(\cdot)$ is discussed for $1\le \rho \le 2$.

Domaines: Analyse fonctionnelle [math.FA]

Catalin Badea, Michel Crouzeix. Numerical radius and distance to unitaries. 2011. ⟨hal-00634780v1⟩