Hankel Operators and Their Applications pp 61-85 | Cite as

# Vectorial Hankel Operators

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## 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 Ω_{j} 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.