A rich structure related to the construction of analytic matrix functions

We study certain interpolation problems for analytic 2 × 2 matrix-valued functions on the unit disc. We obtain a new solvability criterion for one such problem, a special case of the µ-synthesis problem from robust control theory. For certain domains X in C² and C³ we describe a rich structure of interconnections between four objects: the set of analytic functions from the disc into X , the 2 × 2 matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. This rich structure combines with the classical realisation formula and Hilbert space models in the sense of Agler to give an effective method for the construction of the required interpolating functions.