Studying analytic functions in two variables

Integral representations of analytic functions are one of the central tools of single-variable complex and functional analysis. For example, analytic functions \(f\) that map the unit disk into the right complex half plane with \(f(0) = 1\) all possess a Herglotz representation

$$ f(z) = \int_0^{2\pi} \frac{1 + e^{-i\theta}z}{1 - e^{-i\theta}z} \, d\mu(\theta), $$

where the behavior of \(f\) is encoded in the measure \(\mu\). Integral representations can be used to probe the behavior of analytic functions on the boundary of their domains.

This talk provides an introduction to the operator-theoretic view of integral representations and the natural way this view generalizes to two or more variables, including Jim Agler’s Hilbert space model equation and transfer function realization, which encode function behavior into the structure of linear operators on a Hilbert space. We see an application to boundary behavior of functions in the Schur class, and finish with open questions amenable to the Hilbert space approach.

Further reading, by theme. Realizations and models: Agler, “On the representation of certain holomorphic functions defined on a polydisk”; Agler–Tully-Doyle–Young, “Nevanlinna representations in several variables”; Pascoe–Tully-Doyle, “Free Pick functions”; Bickel, “Fundamental Agler decompositions”; Bickel–Knese, “Canonical Agler decompositions and transfer function realizations” and “Inner functions on the bidisk”. Probability and mathematical physics: Cima–Matheson–Ross, “The Cauchy Transform”; Agler–McCarthy, “Hankel vector moment sequences”; Liaw–Treil, “Singular integrals, rank one perturbations, and Clark measure”; Ross, “Lens lectures on Aleksandrov–Clark measures”; Pascoe–Sargent–Tully-Doyle, “A tangential Julia–Carathéodory theory”. Boundary behavior: Agler–McCarthy–Young, “A Carathéodory theorem for the bidisk” and “Operator monotone functions and Löwner’s theorem in several variables”; Tully-Doyle, “Analytic functions on the bidisk at boundary singularities”; Bolotnikov–Kheifits, “The higher order Carathéodory–Julia theorem”; McCarthy–Pascoe, “The Julia–Carathéodory theorem revisited”; Pascoe, “An inductive Julia–Carathéodory theorem for Pick functions in two variables”. Non-commutative theory: Helton–Klep–McCullough, “Proper free analytic maps”. Rational inner functions: the Bickel–Pascoe–Sola sequence of papers, and Knese’s work on stable polynomials.