A+ CATEGORY SCIENTIFIC UNIT

A commutant lifting theorem on analytic polyhedra

Volume 67 / 2005

Calin Ambrozie, Jörg Eschmeier 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.

Authors

  • Calin AmbrozieInstitute of Mathematics,
    Romanian Academy
    PO Box 1-764,
    70700 Bucharest,
    Romania
    e-mail
  • Jörg EschmeierFachrichtung Mathematik,
    Universität des Saarlandes
    Postfach 151150,
    D-66041 Saarbrücken,
    Germany
    e-mail

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