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-to-one” 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 criteria 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 multi-analytic operator.