An Mq(T)-functional calculus for power-bounded operators on certain UMD spaces

For 1 q 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the directionq > 1. More- over, when taken in conjunction with vector-valued transference, this Mq(T)-multiplier result shows that if X2I, and U is an invertible power-bounded operator on X, then U has an Mq(T)-functional calculus for an appropriate range of values of q > 1. The class I includes, in particular, all closed subspaces of the von Neumann{Schatten p-classesCp (1 < p