Abstract

Given a complex manifold M endowed with a hermitian metric g and supporting a smooth probability measure μ, there is a naturally associated Dirichlet form operator A on L²(μ). If b is a function in L²(μ) there is a naturally associated Hankel operator H_b defined in holomorphic function spaces over M. We establish a relation between hypercontractivity properties of the semigroup e^−tA and boundedness, compactness and trace ideal properties of the Hankel operator H_b. Moreover there is a natural algebra R of holomorphic functions on M, analogous to the algebra of holomorphic polynomials on C^m, and which is determined by the spectral subspaces of A. We explore the relation between the algebra R and the Hilbert-Schmidt character of the Hankel operator H_b. We also show that the reproducing kernel is very well related to the operator A.

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