Revista Matemática Iberoamericana

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Volume 3, Issue 1, 1987, pp. 61–138
DOI: 10.4171/RMI/46

Hankel Forms and the Fock Space

Svante Janson[1], Jaak Peetre[2] and Richard Rochberg[3]

(1) Department of Mathematics, Uppsala Universitet, P.O. Box 480, SE-751 06, UPPSALA, SWEDEN
(2) Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00, LUND, SWEDEN
(3) Department of Mathematics, Washington University, Cupples 1, Campus Box 1146, MO 63130-4899, ST. LOUIS, UNITED STATES

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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Janson Svante, Peetre Jaak, Rochberg Richard: Hankel Forms and the Fock Space. Rev. Mat. Iberoamericana 3 (1987), 61-138. doi: 10.4171/RMI/46