Two-parameter lifting theorems and double Hilbert transforms in commutative and non-commutative settings

Abstract

The Nagy-Foias lifting theorem, viewed as an extension property of forms invariant under a 1-parameter unitary group, is generalized to the case of two such groups, which commute either in the ordinary or in the Weyl sense. For the spaces L 2( T 2) and its quantized analogues- L 2( L 2 R 2) and L 2 A 2 where A is the dual of the Heisenberg group—a generalized Bochner theorem (GBT) is also given, providing integral representation of the forms. In particular, corresponding versions of the Nehari theorem are obtained. An abstract analogue of the double Hilbert transform is introduced, and the GBT gives characterizations of the weights for which this operator is continuous. In the case of the torus, this includes characterizations of pairs of weights for the double Hilbert transform.