Failure of Nehari's theorem for multiplicative Hankel forms in Schatten classes

Ole Fredrik Brevig¹ ; Karl-Mikael Perfekt¹

¹Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) NO-7491 Trondheim, Norway

Studia Mathematica, Tome 228 (2015) no. 2, pp. 101-108

Résumé

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.

Détail

DOI : 10.4064/sm228-2-1

Keywords: ortega cerd seip demonstrated there bounded multiplicative hankel forms which arise bounded symbols other form hilbert schmidt class mathcal helson showed has bounded symbol present work investigates forms belonging schatten classes between these cases shown every log log there exist multiplicative hankel forms schatten class mathcal which lack bounded symbols lower bound certain sense optimal symbol multiplicative hankel form product homogeneous linear polynomials