Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes

TL;DR

This paper explores the limitations of Nehari's theorem for multiplicative Hankel forms within Schatten classes, revealing that for certain p-values, forms exist without bounded symbols, extending previous results.

Contribution

It demonstrates the failure of Nehari's theorem for multiplicative Hankel forms in Schatten classes beyond the Hilbert-Schmidt case, identifying a specific p-range where bounded symbols do not exist.

Findings

Existence of multiplicative Hankel forms in Schatten classes p with no bounded symbols.

The lower bound on p for the non-existence of bounded symbols is shown to be optimal in certain cases.

The results extend the understanding of symbol boundedness in relation to Schatten class membership.

Abstract

Ortega-Cerd\`a -- Seip demonstrated that there are bounded multiplicative Hankel forms which do not arise from bounded symbols. On the other hand, when such a form is in the Hilbert-Schmidt class $\mathcal{S}_2$, Helson showed that it has a bounded symbol. The present work investigates forms belonging to the Schatten classes between these two cases. It is shown that for every $p>(1-\log{\pi}/\log{4})^{-1}$ there exist multiplicative Hankel forms in the Schatten class $\mathcal{S}_p$ which lack bounded symbols. The lower bound on $p$ is in a certain sense optimal when the symbol of the multiplicative Hankel form is a product of homogeneous linear polynomials.