# Hankel Forms and the Fock Space

### Svante Janson

Uppsala Universitet, Sweden### Jaak Peetre

Lund University, Sweden### Richard Rochberg

Washington University, St. Louis, USA

## 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.

## 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