Vol. 205, No. 1, 2002

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Hankel operators over complex manifolds

Thomas Deck and Leonard Gross

Vol. 205 (2002), No. 1, 43–97

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 etA and boundedness, compactness and trace ideal properties of the Hankel operator Hb. Moreover there is a natural algebra of holomorphic functions on M, analogous to the algebra of holomorphic polynomials on m, and which is determined by the spectral subspaces of A. We explore the relation between the algebra 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.

Received: 22 May 2000
Revised: 7 May 2001
Published: 1 July 2002
Thomas Deck
Fakultät für Mathematik und Informatik
Universität Mannheim
D-68131 Mannheim, Germany
Leonard Gross
Department of Mathematics
Cornell University
Ithaca, NY 14853