Hilbert points in Hilbert space-valued $L^p$ 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 $\phi$ such that $\Vert \phi {\Vert }_p\le \Vert \phi +f{\Vert }_p$ whenever $⟨f,\phi ⟩=0$. We demonstrate that $\phi$ is a Hilbert point in $L^p(\Omega ;H)$ for some $p\ne 2$ if and only if $\Vert \phi (\omega ){\Vert }_H$ 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.

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