Dilation Theory of Contractions and Nevanlinna-Pick Interpolation Problem

Abstract

In this article, we give two different proofs of the existence of the minimal isometric dilation of a single contraction. Then using the existence of a unitary dilation of a contraction, we prove the `von Neumann's inequality'. Next we give a complete description of the dilation of a pure contraction. We also discuss Ando's proof of the existence of a unitary dilation of a pair of commuting contractions and give an example to show that this result does not hold, in general, for more than two commuting contractions. Then we describe and prove the `commutant lifting theorem' and lastly, we use this theorem to prove the operator valued `Nevanlinna-Pick interpolation problem'.