A Commutant Lifting Theorem for a Domain in C2and Spectral Interpolation

We characterise the pairs of commuting operators on Hilbert space for which the domainΓ={(λ1+λ2, λ1λ2): |λ1|⩽1, |λ2|⩽1} is a complete spectral set. We give an application to the spectral Nevanlinna–Pick problem: we obtain a necessary condition for the existence of an analytic 2×2 matrix function satisfying interpolation conditions and bounds on eigenvalues.