ABSTRACT
Quantum analogues of the notion of Hilbert transform and that of Hankel forms can be formulated by replacing the ordinary Fourier operator in L2 by the Weyl or the Bargman-Weyl transforms. Quadratic inequalities for Hilbert operators or Hankel forms were studied in through a general lifting theorem closely related to classical theorems of Nagy-Foias and Bergman-Shiffer, and through representations of the liftings by operators in the Fock space. This chapter summarizes those results and points out another variant based on the Martineau-Aizenberg representation of functionals in spaces A(D) of analytic functions, which applies also to L2(B), for B the Bergman or other reproducing kernel spaces.
