Hankel Operators over the Complex Wiener Space

Abstract

This work introduces and investigates (small) Hankel operators H b on the Hilbert space of holomorphic, square integrable Wiener functionals. A regularity condition on the symbol b, which guarantees the boundedness of H b , is provided. The symbols b for which H b is of Hilbert–Schmidt type are characterized, and a representation of H b by an integral operator is given. The proofs employ the hypercontractivity of the Ornstein–Uhlenbeck semigroup, together with approximations by finitely many variables. These results extend known results from a finite-dimensional context.