We consider Hankel forms on the Hilbert space of analytic functions square integrable with respect to a given measure on a domain in . Under rather restrictive hypotheses, essentially implying «homogeneity» of the set-up, we obtain necessary and sufficient conditionsfor boundedness, compactness and belonging to Schatten classes , 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 -spaces.
Our theory applies in particular to the Fock spaces (defined by a Gaussian measure in ). For the corresponding -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 by and by , where 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 . In this context we introduce some new function spaces , 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. 61–138DOI 10.4171/RMI/46