Abstract: 

In this note a commutant lifting theorem for
vector-valued functional Hilbert spaces over generalized
analytic polyhedra in $\Bbb{C}^n$ is proved. Let $T$ be
the compression of the multiplication tuple $M_z$ to
a $*$-invariant closed subspace of the underlying functional
Hilbert space.
Our main result characterizes those operators in the
commutant of $T$ which possess a lifting to a
multiplier with Schur class symbol. As an application we obtain
interpolation results of Nevanlinna-Pick and
Carath\'{e}odory-Fej\'{e}r type for Schur class functions.
Our methods apply in particular to the unit ball, the unit polydisc
and the classical symmetric domains of types I, II and III.

1. Institute of Mathematics,
   Romanian Academy
   PO Box 1-764,
   70700 Bucharest,
   Romania
   
2. Fachrichtung Mathematik,
   Universität des Saarlandes
   Postfach 151150,
   D-66041 Saarbrücken,
   Germany