Hankel Forms and the Fock Space

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Abstract
We consider Hankel forms on the Hilbert space of analytic functions square integrable with respect to a given measure on a domain in \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 S_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 L^p -spaces.Our theory applies in particular to the Fock spaces (defined by a Gaussian measure in \mathbb C^n ). For the corresponding L^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 f by f\phi and d\mu by |\phi|^{–2} d\mu , where \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 (- y^2)dx \; dy . In this context we introduce some new function spaces E_p , which are Fourier, and even Mehler invariant.

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