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

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