The Schur class, denoted by S(D), is the set of all functions analytic and bounded by one in modulus in the open unit disc D in the complex plane C, that isThe elements of S(D) are called Schur functions. Suppose ϕ : D → C is a function. A classical result going back to I. Schur states: ϕ ∈ S(D) if and only if there exist a Hilbert space H and an isometry (known as colligation matrix or scattering matrix and nonunique in general)such that ϕ admits a transfer function realization corresponding to V , that isAn analogous statement holds true for Schur functions on the bidisc. On the other hand, Schur-Agler class functions in several variables (a class of analytic functions of more than one variable on the unit ball and more than two variables on the unit polydisc in C n ) is a well-known "analogue" of Schur functions on D. In this paper, we present algorithms to factorize Schur functions and Schur-Agler class functions in terms of colligation matrices. More precisely, we isolate checkable conditions on colligation matrices that ensure the existence of Schur (Schur-Agler class) factors of a Schur (Schur-Agler class) function and vice versa. This paper also seeks to contribute to the understanding of the delicate structure of bounded analytic functions in several complex variables.
In this paper, motivated by the Berger, Coburn and Lebow and Bercovici, Douglas and Foias theory for tuples of commuting isometries, we study analytic representations and joint invariant subspaces of a class of commuting n-isometries and prove that the C *algebra generated by the n-shift restricted to an invariant subspace of finite codimension in H 2 (D n ) is unitarily equivalent to the C * -algebra generated by the n-shift on H 2 (D n ).
Let H 2 (D n ) denote the Hardy space over the polydiscWe present two applications: first, we obtain a dilation theorem for Brehmer n-tuples of commuting contractions, and, second, we relate joint invariant subspaces with factorizations of inner functions. All results work equally well for general vector-valued Hardy spaces. Contents 1. Introduction 1 2. Proof of Theorem 1.1 4 3. Isometric dilations 9 4. Factorizations and invariant subspaces 10 References 14
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