Hilbert points in Hilbert space-valued Lᵖ spaces

Abstract

Let \(H\) be a Hilbert space and \((\Omega,\mathcal{F},\mu)\) a probability space. A Hilbert point in \(L^p(\Omega; H)\) is a nontrivial function \(\varphi\) such that \(\|\varphi\|_p \leq \|\varphi+f\|_p\) whenever \(\langle f, \varphi \rangle = 0\). We demonstrate that \(\varphi\) is a Hilbert point in \(L^p(\Omega; H)\) for some \(p\neq2\) if and only if \(\|\varphi(\omega)\|_H\) assumes only the two values \(0\) and \(C>0\). We also obtain a geometric description of when a sum of independent Rademacher variables is a Hilbert point.