Norm attaining vectors and Hilbert points

Abstract

Let H be a Hilbert space that can be embedded as a dense subspace of a Banach space X such that the norm of the embedding is equal to 1. We consider the following statements for a nonzero vector φ in H: (A) ∥φ∥X =∥φ∥H. (H) ∥φ+f∥X ≥∥φ∥X for every f in H such that ⟨f,φ⟩ = 0. We use duality arguments to establish that (A) =⇒ (H), before turning our attention to the special case when the Hilbert space in question is the Hardy space H2(Td) and the Banach space is either the Hardy space H1(Td) or the weak product space H2(Td)⊙H2(Td). If d = 1, then the two Banach spaces are equal and it is known that (H) =⇒ (A). If d ≥ 2, then the Banach spaces do not coincide and a case study of the polynomials φα(z) = z2 1 +αz1z2 +z2 2 for α ≥ 0 illustrates that the statements (A) and (H) for the two Banach spaces describe four distinct sets of functions.