Journal of Operator Theory

Volume 50, Issue 1, Summer 2003 pp. 179-208.

A tensor product approach to the operator corona problem

Authors: Pascale Vitse
Author institution: Departement de Mathematiques et Statistiques, Universite Laval, Quebec G1K 7P4 QC, Canada

Summary: Let $F$ be a bounded analytic function on the unit disc $\D$ having values in the space $L({\cal H})$ of bounded operators on a Hilbert space ${\cal H}$. The Operator Corona Problem is to decide whether the existence of a uniformly bounded family of left inverses of $F(z)$, $z \in \D$, guarantees the existence of a bounded analytic left inverse of $F$. When $\cal H$ is infinite dimensional, in general, the answer is known to be negative. Some sufficient conditions on values and/or functional properties of $F$ are given for the answer to be positive. The technique uses the tensor product slicing method and the Grothendieck Approximation Property.

Keywords: Operator Corona problem, Bézout equations, tensor product, slicing, Grothendieck Approximation Property, Nevanlinna type meromorphic pseudocontinuation, Sz.-Nagy-Foiaș model spaces

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