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

Volume 228 / 2015

Ole Fredrik Brevig, Karl-Mikael Perfekt 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.

Authors

  • Ole Fredrik BrevigDepartment of Mathematical Sciences
    Norwegian University of Science and Technology (NTNU)
    NO-7491 Trondheim, Norway
    e-mail
  • Karl-Mikael PerfektDepartment of Mathematical Sciences
    Norwegian University of Science and Technology (NTNU)
    NO-7491 Trondheim, Norway
    e-mail

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