Hankel forms and the Fock space
The paper develops in a systematic way a theory of Hankel forms over quite general domains in \({\mathbb{C}}^ n\). Namely, a domain \(\Omega \subset {\mathbb{C}}^ n\) and an absolutely continuous measure \(\mu\) on \(\Omega\) with continuous, strictly positive Radon-Nikodym derivative \(d\mu\) /dm (m is the Lebesgue measure) ar given. The basic Hilbert space \(A^ 2(\mu)={\mathcal H}ol(\Omega)\cap L^ 2(\mu)\) of square integrable holomorphic functions has the reproducing kernel K(z,w). The associated measure \(\nu\) is defined as \(d\nu (z)=d\mu /K(z,z).\) We put L for the reproducing kernel in the Hilbert space \(A^ 2(\nu)\) and Q for the orthogonal projection Q: \(L^ 2(\nu)\to A^ 2(\nu)\). The whole theory of Hankel forms on \(A^ 2(\mu)\) is developed under the following basic assumptions: V1. \(K(z,z)>0\), \(\forall z\in \Omega,\) V2. \(L(z,w)=\kappa (K(z,w))^ 2\) for some constant \(\kappa\geq 0,\) V3. If \(f\in {\mathcal H}ol(\Omega)\) and \(\int | L(z,w)f(w)| d\nu (w)<\infty\) for every z, then \[ \int L(z,w)f(w)d\nu (w)=f(z),\quad z\in \Omega. \] For a function b on \(\Omega\) the Hankel forms \(H_ b\) and \(\Gamma_ b\) on \(A^ 2(\mu)\) (resp. on \(A^ 2(\nu))\) are given by \[ H_ b(f,g)=\int_{\Omega}\bar bfg d\mu. \] Before stating the main result of the first part of the paper ({\S}{\S} 1-6) recall one more concept: the natural scale of weighted \(A^ p\)-spaces. Set \(\omega (z)=K(z,z)^{-1}\). An analytic function f is in \(A^ p_{\omega}\) iff \[ \int_{\Omega}| f|^ p \omega^{p-2}d\nu <\infty \] (f is in \(A^{\infty}_{\omega}\) iff \(\omega\) f is essentially bounded on \(\Omega\)). Let \(\alpha^{\infty}_{\omega}\) be the closure of \(A^ 2_{\omega}\) in the \(A^{\infty}_{\omega}\)-metric. The main result reads as follows. Theorem A. Under the assumptions V1-V3 we have (a) \(\Gamma_ b\) is bounded (in \(A^ 2(\mu))\) if and only if \(Qb\in A^{\infty}_{\omega},\) (b) \(\Gamma_ b\) is compact if and only if \(Qb\in a^{\infty}_{\omega},\) (c) \(\Gamma_ b\) is in \(S_ p\), where \(1\leq p<\infty\) if and only if \(Qb\in A^ p_{\omega}.\) The second part of the paper is devoted to the study of the Fock spaces \(F^ p_{\infty}\) and Hankel forms on them. Let \(d\mu_{\alpha}=(\alpha /\pi)^ ne^{-\alpha | z|^ 2}dm(z)\), \(\alpha >0\) be the family of measures on \({\mathbb{C}}^ n\). Then \(F^ p_{\alpha}\) (1\(\leq p\leq \infty)\) is the space of entire functions f such that \(f(z)e^{-\alpha | z|^ 2}\in L^ p(m)\). For Hankel forms on \(F^ 2_{\alpha}\) the analogue of Theorem A is shown (by applying the earlier results of the paper). The relationship between operator theoretic properties of the Hankel form, given for an entire function b by \[ H_ b^{2\alpha}(f,g)=\int_{{\mathbb{C}}^ n}\bar bfg d\mu_{2\alpha}, \] and function properties of b, are illustrated by the following result. Theorem B. Let \(K^{\alpha}_ w(z)=e^{\alpha <z,w>-\alpha | w|^ 2/2}.\) The following are equivalent for \(1\leq p\leq \infty:\) (i) \(H_ b^{2\alpha}\in S_ p.\) (ii) \(H_ b^{2\alpha}(f,g)=\sum \lambda_ i<f,k^{\alpha}_ i><g,k^{\alpha}_ i>\) where \(\{\lambda_ i\}\in \ell^ p\) (and \(k^{\alpha}_ i=k^{\alpha}_{wi}\) for a suitable sequence \(\{w_ i\}\), separation condition etc.). (iii) \(b=\sum \lambda_ ik_ i^{2\alpha}\) with \(\{\lambda_ i\}\in \ell^ p\) (same qualification for \(k_ i^{2\alpha}).\) (iv) \(b\in P_{2\alpha}(L^ p_{2\alpha}).\) (v) \(\{d_ n\}^{\infty}_ 0\in \ell^ p\) where \(d_ N\) is the distance in \(F^{\infty}_{2\alpha}\) between b and the set \(P_ N\) of linear combinations \(\sum^{N}_{j=1}a_ jk_ j^{2\alpha}\) of length N. (vi) \(b\in F^ p_{2\alpha}.\) The whole theory is invariant under the so called ``gauge transformations'' \(f\to \phi f\), \(\mu \to | \phi |^{-2}\mu\), where \(\phi\) is analytic and non-zero function in \(\Omega\). This allows an interesting alternative interpretation of the obtained results. The paper contains many other valuable results which we do not describe here. It surely deserves attention of specialists working in function and/or operator theory.
scientific article; zbMATH DE number 4155253