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

• 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).
Mathematics Subject Classification: Primary: 47A25; Secondary: 47A20, 15A60.

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