Completely bounded transformations of H-operator spaces

AbstractIf U is the commutant of a strictly cyclic unilateral weighted shift with a monotonically decreasing weight sequence, then we show that there is a natural isomorphism of the Banach space of bounded linear maps from U into B(H) with the Banach space of bounded linear maps of the trace class operators into H, where H is a separable, infinite dimensional Hilbert space. Under this isomorphism, an operator Φ from U into B(H) is completely bounded if and only if its image extends to a bounded map of the Hilbert-Schmidt operators into H. The proof shows that if Φ is only completely row bounded, then Φ is in fact completely bounded. The characterization of the completely bounded maps is then used to prove the existence of a family of completely unbounded representations of U into B(H)