Uniqueness of invariant Hahn-Banach extensions

Abstract

Let 𝓁 be a linear functional on a subspace 𝑌 of a real linear space 𝑋 provided with a sublinear functional 𝑝 with 𝓁 ≤ 𝑝 on 𝑌. If 𝒢 is an abelian semigroup of linear transformations 𝑇: 𝑋 →𝑋 such that 𝑇(𝑌 ) ⊆ 𝑌 𝑝 (𝑇𝓍) ≤ 𝑝 (𝓍) and 𝓁(𝑇𝓎) = 𝓁(𝓎) for all 𝑇 ∊ 𝒢, 𝓍 ∊ 𝑋 and 𝓎 ∊ 𝑌 respectively, then a generalization of the classical Hahn-Banach theorem asserts that there exists an extension 𝓁 of 𝓁, 𝓁 ≤ 𝑝 on 𝑋 and 𝓁` remains invariant under 𝒢. The present paper investigates various equivalent conditions for the uniqueness of such extensions and these are related to nested sequences of 𝑝-balls, a concept that has proved useful in recent years in dealing with such extensions. The results are illustrated by a variety of examples and applications.