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 $1$. We consider the following statements for a nonzero vector $\varphi $ in $H$:
(A) $\|\varphi \|_X = \|\varphi \|_H$.
(H) $\|\varphi +f\|_X \geq \|\varphi \|_X$ for every $f$ in $H$ such that $\langle f, \varphi \rangle =0$.
We use duality arguments to establish that (A)$\Rightarrow $(H), before turning our attention to the special case when the Hilbert space in question is the Hardy space $H^2(\mathbb T^d)$ and the Banach space is either the Hardy space $H^1(\mathbb T^d)$ or the weak product space $H^2(\mathbb T^d) \odot H^2(\mathbb T^d)$. If $d=1$, then the two Banach spaces are equal and it is known that (H)$\Rightarrow $(A). If $d\geq 2$, then the Banach spaces do not coincide and a case study of the polynomials $\varphi _\alpha (z) = z_1^2 + \alpha z_1 z_2 + z_2^2$ for $\alpha \geq 0$ illustrates that the statements (A) and (H) for these two Banach spaces describe four distinct sets of functions.
