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 Hb 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
Hb. Moreover there is a natural algebra ℛ of holomorphic functions on M,
analogous to the algebra of holomorphic polynomials on ℂm, 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 Hb. We
also show that the reproducing kernel is very well related to the operator
A.
