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HANKEL OPERATORS OVER COMPLEX MANIFOLDS
 PACIFIC JOURNAL OF MATHEMATICS
, 2002
"... Given a complex manifold M endowed with a hermitian metric g andsupporting a smooth probability measure µ, there is a naturally associatedDirichlet form operator A on L2 (µ). If b is a function in L2 (µ) there is a naturally associatedHankel operator Hb defined in holomorphic function spaces over M. ..."
Abstract

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Given a complex manifold M endowed with a hermitian metric g andsupporting a smooth probability measure µ, there is a naturally associatedDirichlet form operator A on L2 (µ). If b is a function in L2 (µ) there is a naturally associatedHankel operator Hb defined in holomorphic function spaces over M. We establish a relation between hypercontractivity properties of the semigroup e−tA and boundedness, compactness andtrace 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, andwhich is determinedby the spectral subspaces of A. We explore the relation between the algebra R andthe HilbertSchmidt character of the Hankel operator Hb. We also show that the reproducing kernel is very well relatedto the operator A.