i ¢ on a Hilbert space whose inner product is defined in terms of periodic distributions and we relate this model theory for the case when m = 2 to a disconjugacy theory for a subclass of Toeplitz operators of the type studied by Boutet de Monvel and Guilliman, classical function theoretic ideas on the Dirichlet space, and the theory of nonstationary stochastic processes. This is presented in a series of three papers. In this first paper, we concentrate on a model for these T.

A toral algebraic set A is an algebraic set in C n whose intersection with T n is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.

In part 1 and 2 of this series [AgSt1], [AgSt2], an operator T was defined to be an m-isometry if ~ (-1) k('J:) (T*)m-k Tm-k = O. A model for m-isometric operatorsk=O was given as multiplication by e i ¢ on a Hilbert space whose inner product is defined in terms of periodic distributions. We related this model theory for the case when m = 2 to a disconjugacy theory for a subclass of Toeplitz operators of the type studied by Boutet de Monvel and Guillemin, classical function theoretic ideas on the Dirichlet space, and the theory of nonstationary stochastic processes.This third paper of the series consists of Sections 8,9 and 10 and turns to the study of 2-isometries possessing a cyclic vector and gives some concluding remarks and presents a list of open questions related to all three parts of the paper.Section 8 is devoted to the study of 2-isometries possessing a cyclic vector. It turns out that if T is a 2-isometry possessing a cyclic vector, then dim ker (IIT*T ~ 111-(T*T -1)) = 1 and furthermore if fa i-0 is chosen so thatthen fa is cyclic for T if and only if T is pure. Here, a 2-isometry T is said to be pure if it has no isometric direct summand. If one uses the cyclic vector fa, then one can construct a model for T based on a probability measure on the unit circle. This model leads to a rich function theoretic analysis of 2-isometries.Section 9 derives a number of qualitative facts about the existence and uniqueness of certain types of extensions, so called Brownian unitary and isometric extensions, of a 2-isometry possessing a cyclic vector.

In the first part of this series [AgSt], a model for operators satisfying the space whosc inner product is defined in terms of periodic distributions. In this paper and the next, wc relate this model theory for the ca.'3e when m = 2 to a disconjugacy theory for a subclass of Toeplitz operators of the type studied by Boutet de Monvel and Guillemin, cla.ssical function theoretic ideas on the Dirichlet space, and the theory of nonstationary stochastic processes.

MSC: 47A62 47A05 47B99 Keywords: Hilbert Space m-Isometric operator n-Symmetric operator Elementary operators Nilpotent operators Tensor products Linear transformationsWe study the sum of an m-isometry or an m-symmetric operator with a nilpotent operator. We also study the product of two m-isometries or two n-symmetries. We obtain several theorems generalizing previous work. Our method is straightforward and thus offers direct and simple insights into why these theorems are true.We will use the hereditary functional calculus which was developed by J. Agler [1] and motivated by the work of W.J. Helton and J. Ball [12,7]. Let H be a Hilbert space and L(H) be the set of all bounded linear operators on H. For a polynomial p(x, y) in two variables x and y, p(x, y) = m k=0 m =0 c k y x k , c k ∈ C