Hankel Operators over Complex Manifolds

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(μ). If b is a function in L(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 C, 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.