Operator positivity and analytic models of commuting tuples of operators

We study analytic models of operators of class C-.0 with natural positivity assumptions. In particular, we prove that for an m-hypercontraction T is an element of C-.0 on a Hilbert space H, there exist Hilbert spaces epsilon and epsilon(*) and a partially isometric multiplier theta is an element of M(H-2 (epsilon), A(m)(2) (epsilon(*))) such that H congruent to Q(theta) - A(m)(2) (epsilon*) circle minus theta H-2(epsilon) and P-Q theta M-z vertical bar Q(theta), where A(m)(2) (epsilon(*)) is the epsilon(*)-valued weighted Bergman space and H-2 (epsilon) is the E-valued Hardy space over the unit disc a We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their applications to joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over the polydisc. In particular, we completely analyze doubly commuting quotient modules of a large class of reproducing kernel Hilbert modules, in the sense of Arazy and Englis, over the unit polydisc D-n.