An Operator-valued Berezin Transform and the Class of n-Hypercontractions

Abstract.

We study an operator-valued Berezin transform corresponding to certain standard weighted Bergman spaces of square integrable analytic functions in the unit disc. The study of this operator-valued Berezin transform relates in a natural way to the study of the class of n-hypercontractions on Hilbert space introduced by Agler. To an n-hypercontraction $T \in {\mathcal{L}}({\mathcal{H}})$ we associate a positive ${\mathcal{L}}(\mathcal{H})$-valued operator measure dω _{n, T} supported on the closed unit disc $\bar{\mathbb{D}}$ in a way that generalizes the above notion of operator-valued Berezin transform. This construction of positive operator measures dω _{n, T} gives a natural functional calculus for the class of n-hypercontractions. We revisit also the operator model theory for the class of n-hypercontractions. The new results here concern certain canonical features of the theory. The operator model theory for the class of n-hypercontractions gives information about the structure of the positive operator measures dω _{n, T} .