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Vectorial Hankel Operators

  • Vladimir Peller
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we study Hankel operators on spaces of vector functions. We prove in §2 a generalization of the Nehari theorem which describes the bounded block Hankel matrices of the form \( {\left\{ {{\Omega _{j}}_{{ + k}}} \right\}_{{j,k}}} \geqslant 0\), where the Ωj are bounded linear operators from a Hilbert space H to another Hilbert space K. The proof is based on a more general result on completing matrix contractions. This result is obtained in §1. Namely, we obtain in §1 a necessary and sufficient condition on Hilbert space operators A, B, and C for the ex- istence a Hilbert space operator Z such that the block matrix \( \left( {\begin{array}{*{20}{c}} A & B \\ C & Z \\\end{array} } \right)\) is a contraction (i.e., has norm at most 1). Moreover, we describe in §1 all solutions of this completion problem. Note that the results of §2 will be used in Chapter 5 to parametrize all solutions of the Nehari problem.

Keywords

Hilbert Space Bounded Linear Operator Scalar Case Finite Rank Carleson Measure 
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Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • Vladimir Peller
    • 1
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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