Parametrization of Solutions of the Nehari Problem

Abstract

For a Hankel operator Γ from H² to H² 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 φ₀ 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 φ₀ satisfying ||φ₀ — 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.