Hankel Forms and the Fock Space

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 $S_p,p\ge 1$, 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 $f\varphi$ and $d\mu$ by $|\varphi {|}^{–2}d\mu$, where $\varphi$ 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 $(-y^2)dxdy$. In this context we introduce some new function spaces $E_p$, which are Fourier, and even Mehler invariant.