8 - Models for ω-Hypercontractive Operator Tuples

Summary

Chapter 8 presents a de Branges–Rovnyak-type model theory for a given operator-tuple in an appropriate class (indexed by an admissible weight ?) of hypercontractive operator tuples. Application of results from Chapter 4 leads to a shift-type model-operator-tuple acting on a backward-shift-invariant contractively included subspace of a ?-weighted Hardy–Fock space. In the nicest case one can use results of Chapter 5 to define a characteristic operator function from which one can recover in the model the original ?-hypercontractive operator tuple up to unitary equivalence. A particular instance of this nice situation is the case where the ?-hypercontractive operator tuple is pure, or equivalently, when the backward-shift-invariant subspace is isometrically included in the ambient weighted Hardy–Fock space. In this case, there is also an alternative model theory based on a characteristic Bergman-inner family which makes use of results from Chapter 7. In this case, there is also a Bergman-inner characteristic function which is a partial unitary invariant and relates to the work on the Bergman shift from the 1990s.