A commutant lifting theorem on analytic polyhedra
Volume 67 / 2005
Banach Center Publications 67 (2005), 83-108
MSC: Primary 47A57; Secondary
47A13, 47A20, 41A05.
DOI: 10.4064/bc67-0-7
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 Carathéodory-Fejér 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.
