In this paper we discuss compactness of the canonical solution operator to ∂ on weigthed L 2 spaces on C n . For this purpose we apply ideas which were used for the Witten Laplacian in the real case and various methods of spectral theory of these operators. We also point out connections to the theory of Dirac and Pauli operators.2000 Mathematics Subject Classification. Primary 32W05; Secondary 32A36, 35J10, 35P05.

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We study certain densely defined unbounded operators on the Segal-Bargmann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding D-complex and study properties of the corresponding complex Laplacian˜ D = DD * + D * D, where D is a differential operator of polynomial type.

In this paper, we generalize several results about the ∂-complex on the Segal-Bargmann space of C n to weighted Bergman spaces on Hermitian manifolds. We also study in detail the ∂-complex on the unit ball with the complex hyperbolic metric and a non-Kähler metric. The former case turns out to have duality properties similar to the Segal-Bargmann space while the later exhibits a different behavior. We apply these results to solve the ∂-equation on the Bergman spaces in the unit ball of C n with the "exponential" and "standard" weights.