THE 2 × 2 SPECTRAL NEVANLINNA–PICK PROBLEM

A method is presented to construct interpolation functions into the 2 × 2 open spectral unit ball. For the spectral Nevanlinna–Pick problem, these functions are in some sense extremal, and the set of all these interpolation functions is enough to solve any interpolation problem, with solvable finite interpolation data. This fact is used to compute the complex geodesics for the symmetrized bidisc and for the spectral unit ball, and to solve completely the two-point interpolation problem for the two target sets.