JournalsrmiVol. 3, No. 1pp. 61–138

Hankel Forms and the Fock Space

  • Svante Janson

    Uppsala Universitet, Sweden
  • Jaak Peetre

    Lund University, Sweden
  • Richard Rochberg

    Washington University, St. Louis, USA
Hankel Forms and the Fock Space cover
Download PDF

Abstract

We consider Hankel forms on the Hilbert space of analytic functions square integrable with respect to a given measure on a domain in Cn\mathbb C^n. Under rather restrictive hypotheses, essentially implying «homogeneity» of the set-up, we obtain necessary and sufficient conditionsfor boundedness, compactness and belonging to Schatten classes Sp,  p1S_p, \; p ≥ 1, for Hankel forms (analogues of the theorems of Nehari, Hartman and Peller). There are several conceivable notions of «symbol»; choosing the appropriate one, these conditions are expressed in terms of the symbol of the form belonging to certain weighted LpL^p-spaces.
Our theory applies in particular to the Fock spaces (defined by a Gaussian measure in Cn\mathbb C^n). For the corresponding LpL^p-spaces we obtain also a lot of other results: interpolation (pointwise, abstract), approximation, decomposition etc. We also briefly treat Bergman spaces.
A specific feature of our theory is that it is «gauge invariant». (A gauge transformation is the simultaneous replacement of functions ff by fϕf\phi and dμd\mu by ϕ2dμ|\phi|^{–2} d\mu, where ϕ\phi is a given (non-vanishing) function). For instance, in the Fock case, an interesting alternative interpretation of the results is obtained if we pass to the measure exp (y2)dx  dy(- y^2)dx \; dy. In this context we introduce some new function spaces EpE_p, which are Fourier, and even Mehler invariant.

Cite this article

Svante Janson, Jaak Peetre, Richard Rochberg, Hankel Forms and the Fock Space. Rev. Mat. Iberoam. 3 (1987), no. 1, pp. 61–138

DOI 10.4171/RMI/46