An Extension Theorem for Bounded Forms Defined in Relaxed Discrete Algebraic Scattering Systems and the Relaxed Commutant Lifting Theorem

Abstract

The concept of relaxed discrete algebraic scattering system is introduced. For a relaxed discrete algebraic scattering system (G, G ¹, G 2, Γ) and a set {B ₁, B ₂, B ₀} of sesquilinear forms defined in the relaxed discrete algebraic scattering system such that B ₁: G ¹ × G ¹ → ℂ and B ₂: G ² × G ² → ℂ are nonnegative, B ₁(Γg ¹, Γg ¹) ≤ B ₁(g ¹, g ¹) for all g ¹ ∈ G ¹, B ₂(Γg,Γg) = B ₂(g, g) for all g ∈ G, and ∣B ₀(g ¹, g ²)∣ ≤ B ₁(g ¹, g ^{1 1/2})B ₂(g ², g ²)^1/2 for all g ¹ ∈ G ¹ and g ² ∈ G ², a map ф: ΓG ¹ → G ¹ interpolating the system and the forms is considered. An extension theorem for a set {B ₁, B ₂, B ₀} of sesquilinear forms defined in a relaxed discrete algebraic scattering system (G, G ¹, G ², Γ) with interpolant map ф: ΓG ¹ → G ¹ is established. It is shown that the result encompasses the Cotlar-Sadosky extension theorem for bounded forms defined in discrete algebraic scattering systems as well as the Relaxed Commutant Lifting Theorem. Furthermore, the interpolants D in the relaxed lifting problem are obtained in correspondence with the extension forms B in a related extension problem so that D and B determine each other uniquely.