Hankel Operators and Their Applications pp 147-229 | Cite as

# Parametrization of Solutions of the Nehari Problem

## Abstract

For a Hankel operator Γ from *H*^{2} to *H*^{2} and *p* ≥ ||Γ||, we consider in this section the problem of describing all symbols *φ* ∈ *L*^{ ∞ } of Γ (i.e., Γ = H_{ φ }) which satisfy the inequality ||*φ*||∞ ≤ *p.* If φ_{0} is a symbol of F, then as we have seen in §1.1, this problem is equivalent to the problem of finding all approximants *f* ∈ *H*^{ ∞ } to *φ*_{0} satisfying ||*φ*_{0} — f*||*_{ ∞ }. ≤ *p*. This problem is called the Nehari problem. If *p =* ||Γ||, a solution yo of the Nehari problem (i.e., a symbol *φ* of Γ of norm at most *φ* is called *optimal*. If *p >* ||Γ||, the solutions of the Nehari problem are called *suboptimal*)*.* Clearly, the optimal solutions of the Nehari problem are the symbols of F of minimal norm.

## Keywords

Unit Ball Toeplitz Operator Canonical Function Minimal Norm Blaschke Product## Preview

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