Characteristic Functions and Joint Invariant Subspaces

Let T := [T1, . . . , Tn] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a “one-toone” correspondence between the joint invariant subspaces under T1, . . . , Tn, and the regular factorizations of the characteristic function ΘT associated with T . In particular, we prove that there is a non-trivial joint invariant subspace under the operators T1, . . . , Tn, if and only if there is a non-trivial regular factorization of ΘT . We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators. We obtain criterions for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multianalytic operator.