Failure of Nehari's theorem for multiplicative Hankel forms in Schatten classes
Volume 228 / 2015
Studia Mathematica 228 (2015), 101-108
MSC: Primary 47B35; Secondary 30B50.
DOI: 10.4064/sm228-2-1
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.
