Hankel Forms and the Fock Space

  • Svante Janson

    Uppsala Universitet, Sweden
  • Jaak Peetre

    Lund University, Sweden
  • Richard Rochberg

    Washington University, St. Louis, USA


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