January  2009, 8(1): 37-54. doi: 10.3934/cpaa.2009.8.37

Intersections of several disks of the Riemann sphere as $K$-spectral sets

1. 

Laboratoire Paul Painlevé, Bât. M2, UMR CNRS no. 8524, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France, France

2. 

Institut de Recherche Mathématique de Rennes, UMR no. 6625, Université de Rennes 1, Campus de Beaulieu, 35042 RENNES Cedex, France

Received  February 2008 Revised  August 2008 Published  October 2008

We prove that if $n$ closed disks $D_1$,$D_2$,...,$D_n$, of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_1\cap D_2\cap...\cap D_n$ is a complete $K$-spectral set for $A$, with $K\leq n+n(n-1)/\sqrt3$. When $n=2$ and the intersection $X_1\cap X_2$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).
Citation: Catalin Badea, Bernhard Beckermann, Michel Crouzeix. Intersections of several disks of the Riemann sphere as $K$-spectral sets. Communications on Pure & Applied Analysis, 2009, 8 (1) : 37-54. doi: 10.3934/cpaa.2009.8.37
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