A commutant lifting theorem for a domain in C-2 and spectral interpolation

We characterise the pairs of commuting operators on Hilbert space for which the domain Gamma = {(lambda(1) + lambda(2), lambda(1), lambda(2)) :, \lamb da(1)\ less than or equal to,1, \lambda(2)\ less than or equal to 1} is a c omplete spectral set. We give an application to the. spectral Nevanlinna-Pi ck problem: we obtain a necessary condition for the existence of an analyti c 2 x 2 matrix function satisfying interpolation conditions and bounds on e igenvalecs. (C) 1999 Academic Press.