THE QUASI-CANONICAL SOLUTION OPERATOR TO ∂ RESTRICTED TO THE FOCK-SPACE


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Abstract

We consider the solution operator S: ℱμ,(p,q)L2(μ)(p, q) to the ∂-operator restricted to forms with coefficients in ℱμ = {f: f is entire and ƒn |f(z)|2 dμ(z) < ∞}. Here ℱμ,(p,q) denotes (p,q)-forms with coefficients in ℱμ, L2(μ) is the corresponding L2-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to ∂. 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 [Tzi, Tzi] = [T*zi, Tzi] ℱμL2(μ).

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