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We prove that if two closed disks X_1 and X_2 of the Riemann sphere arespectral sets for a bounded linear operator A on a Hilbert space, then theintersection X_1\cap X_2 is a complete (2+2/\sqrt{3})-spectral set for A. Whenthe intersection of X_1 and X_2 is an annulus, this result gives a positiveanswer to a question of A.L. Shields (1974).

K-spectral sets and intersections of disks of the Riemann sphere