THE 2 x 2 Spectral Nevanlinna-Pick Problem

A method is presented to construct interpolation functions intothe 2 x 2 open spectral unit ball. For the spectral Nevanlinna–Pickproblem, these functions are in some sense extremal, and theset of all these interpolation functions is enough to solveany interpolation problem, with solvable finite interpolationdata. This fact is used to compute the complex geodesics forthe symmetrized bidisc and for the spectral unit ball, and tosolve completely the two-point interpolation problem for thetwo target sets.