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

Abstract

Ortega-Cerdà -- 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.

Minor typographical changes. This paper has been accepted for publication in Studia Mathematica

Cited by in corpus (6)

• The multiplicative Hilbert matrix
• Contractive inequalities for Bergman spaces and multiplicative Hankel forms
• On Helson matrices: moment problems, non-negativity, boundedness, and finite rank
• Weak product spaces of Dirichlet series
• Linear functions and duality on the infinite polytorus
• Projecting onto Helson matrices in Schatten classes