UC San Diego

A Pick function is an analytic function which takes the upper half plane to itself. Classically, many mathematicians, (such as Nevanlinna, Lowner, von Neumann, Wigner and many others) studied the boundary behavior of Pick functions. Importantly, Pick functions have good integral representations which were discovered by Nevanlinna in his solution to the Hamburger moment problem. In this thesis, we will use multivariable Nevanlinna representations to understand the multivariable analogue of Pick functions. Recently, Agler and McCarthy developed a rich theory of the boundary behavior for two variable Pick functions. We extend and refine their theory in this thesis. Specically, we show a higher order Julia- Carathéodory type theorem along the lines of two variable Julia-Carathéodory theorem proven by Agler, McCarthy and Young. We show that if a two variable Pick function f has real residues to order 2N -- 1 at infinity and the imaginary part of the remainder between f and this expansion is of order 2N +1; then f has real residues to order 2N and directional residues to order 2N + 1. Furthermore, f has real residues to order 2N + 1 if and only if the 2N + 1-th derivative is given by a polynomial, thus obtaining a two variable analogue of a higher order Julia-Carathéodory type theorem. We also develop machinery to prove the same theorems for points in R². The edge-of- the-wedge theorem is a theorem used to analytically continue a function through the boundary of a domain under certain conditions, which was discovered by the physicist Bogoliubov, but has proven to be of great use in several complex variables. We discuss an analogous phenomenon, which we call the wedge-of-the-edge theorem, which characterizes the boundary values of Pick functions, functions from the polyupperhalf plane into the half plane. We show that Pick functions which have a continuous real-valued extension to a union of two hypercubes with a certain orientation in Rd have good analytic continuation properties. Furthermore, we establish bounds on the behavior of this analytic continuation, which makes normal families arguments accessible on the boundary for Pick functions in several variables. Later, we give an application to the theory of operator monotone functions