A characteristic operator function for the class of n-hypercontractions

Abstract

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C 0 ⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function W n , T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form W n , T , we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function W n , T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.