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 L 2 (μ). If b is a function in L 2 (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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