Integral Equations and Operator Theory 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 func-tions 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 ∈ L(H) we associate a positive L(H)-valued operator measure dωn,T supported on the closed unit disc D ̄ in a way that generalizes the above notion of operator-valued Berezin transform. This construction of positive operator measures dωn,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 dωn,T