Abstract
In this chapter we will show how one can use the commutant lifting theorem to solve several classical interpolation problems. Indeed we will treat the Carathéodory interpolation problem, the Nevanlinna-Pick interpolation problem, a Hankel matrix interpolation problem, an extension problem involving contractions intertwining isometries and a Toeplitz inversion problem. We will also use the commutant lifting theorem to obtain some Corona theorems for analytic Toeplitz operators.
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