A characteristic operator function for the class of n-hypercontractions

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,W-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,W-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,W-T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space. (c) 2006 Elsevier Inc. All rights reserved.