The Quasi-Canonical Solution Operator to ${\bar \partial }$ Restricted to the Fock-Space

Abstract

We consider the solution operator S: ℱ_μ,(p,q) → L ²(μ)_{(p, q)} to the ${\bar \partial }$-operator restricted to forms with coefficients in ℱ_μ = {f: f is entire and ∫_ℂn |f(z)|² dμ(z) < ∞}. Here ℱ_μ,(p,q) denotes (p,q)-forms with coefficients in ℱ_μ, L ²(μ) is the corresponding L ²-space and μ 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 ⊥ ℱ_(p,q) ∀v ∈ 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 $[T_{\overline {z_i } } ,T_{z_i } ] = [T_{z_i }^* ,T_{z_i } ]$: ℱ_μ → L ²(μ).