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Mathematics/Functional Analysis
Numerical radius and distance from unitary operators
Catalin Badea 1, Michel Crouzeix 2
 1: Laboratoire Paul Painlevé (LPP) http://math.univ-lille1.fr/ CNRS : UMR8524 – Université Lille I - Sciences et technologies U.F.R. de Mathématiques 59 655 Villeneuve d'Ascq Cédex France 2: Institut de Recherche Mathématique de Rennes (IRMAR) http://irmar.univ-rennes1.fr/ CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne France
Analyse numérique
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 $1\le \rho \le 2$.
English
2012-01-24

 Project Id Dynop

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 hal-00634780, version 2 http://hal.archives-ouvertes.fr/hal-00634780 oai:hal.archives-ouvertes.fr:hal-00634780 From: Catalin Badea <> Submitted on: Tuesday, 24 January 2012 16:47:55 Updated on: Tuesday, 24 January 2012 16:56:46