The quasi-canonical solution operator to $\bar{\partial}$ restricted to the Fock-space

Schneider, Georg

Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 947-956.

Résumé

We consider the solution operator $S\:\mathcal F_{\mu ,(p,q)}\rightarrow L^2(\mu )_{(p,q)}$ to the $\bar{\partial }$-operator restricted to forms with coefficients in $\mathcal F_{\mu }= \bigl \lbrace f\: f \text{is} \text{entire} \text{and} \int _{\mathbb{C}^n} |f(z)|^2\mathrm{d}\mu (z) \infty \bigr \rbrace $. Here $\mathcal F_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in $\mathcal F_{\mu }$, $L^2(\mu )$ is the corresponding $L^2$-space and $\mu $ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\bar{\partial }$. This solution operator will have the property $Sv\bot \mathcal F_{(p,q)}\, \forall \,v \in \mathcal F_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators $[T_{\bar{z_i}},T_{z_i}]= [T^*_{{z_i}},T_{z_i}]\:\mathcal F_\mu \rightarrow L^2(\mu )$.

Détail

Classification : 32A15, 32W05, 35N15, 47B35
Keywords: Fock-space; Hankel-operator; reproducing kernel