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On the representation and boundary behavior of certain classes of holomorphic functions in several variables
Abstract
This dissertation concerns the investigation of function theoretic properties of certain classes of holomorphic functions in two or more variables by means of operator theoretic methods. Of primary concern will be the Schur class, the class of holomorphic functions from the complex polydisk into the complex unit disk, and the Pick class, the class of holomorphic functions from the complex poly upperhalfplane into the complex upperhalfplane. In more than two variables, our results will concern certain large subclasses of these functions that satisfy an operator theoretic condition analogous to a classical inequality of functions of one variable due to von Neumann [vN51]. These subclasses are typically referred to as the SchurAgler subclass of the Schur functions (introduced in [Agl90]), and the Loewner subclass of the Pick functions (introduced in [AMY12b]. (In one or two variables, these subclasses coincide with the whole class.) These functions are amenable to investigation by means of an operator theoretic construct called a Hilbert space model, introduced in [Agl90], which relates operator theoretic properties with function theoretic behavior. Hilbert space models are associated with and closely related to the notion of a transfer function realization from engineering and control theory [Hel87]. In Chapter 2, we describe a generalization of Hilbert space models for Schur functions on the bidisk that is wellsuited to the investigation of boundary behavior of a function at a class of singular points for the function on the 2torus. We prove that generalized models with certain regularity properties exist at these singularities. We then solve two function theoretic problems. First, we characterize the directional derivatives of a function in the Schur class at a singular point on the torus where a Caratheodory condition holds (following the generalization of the JuliaCarathedory theorem in [AMY12]. Second, we develop a representation theorem for functions in the twovariable Pick class analogous to the Nevanlinna representation theorem characterizing the Cauchy transforms of positive measures on the real line. In Chapter 3, we investigate more closely the structure of the generalized Hilbert space model. We characterize the directional derivatives in terms of a rational function depending on the structure of a positive contraction associated with a generalized model of a given Schur function. We describe classes of generalized models corresponding to different classes of singular points in the boundary for a Schur function in two variables. In Chapter 4, we generalize to several variables the Nevanlinna representation first investigated in Chapter 2. We show that for the Loewner class, there are representation formulae in terms of denselydefined selfadjoint operators on a Hilbert space that classify completely the Loewner class. We identify four types of such representations, and we obtain functiontheoretic conditions that are necessary and sufficient for a given function to possess a representation of each of the four types
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