[Paper Review] An inductive Julia-Caratheodory theorem for Pick functions in two variables

J. E. Pascoe|arXiv (Cornell University)|May 27, 2016

Meromorphic and Entire Functions5 references3 citations

TL;DR

This paper establishes a higher-order Julia-Carathéodory theorem for two-variable Pick functions by introducing an inductive framework linking asymptotic behavior at infinity to moment conditions. It proves that a function in the intermediate Löwner class $\mathcal{L}^{N-}$ belongs to the full Löwner class $\mathcal{L}^{N}$ if and only if its remainder has vanishing imaginary part of order $2N+1$, and shows that real residues to order $2N+1$ occur precisely when the $2N+1$-th derivative is a polynomial, generalizing classical one-variable results to two variables.

ABSTRACT

We study the asymptotic behavior of Pick functions, analytic functions which take the upper half plane to itself. 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.

Motivation & Objective

• To extend the classical Julia-Carathéodory theorem to two-variable Pick functions by developing a higher-order asymptotic theory at infinity.
• To characterize the Löwner classes $\mathcal{L}^N$ and $\mathcal{L}^{N-}$ in terms of residue and moment conditions.
• To establish an inductive criterion for transitioning between $\mathcal{L}^{N-1}$, $\mathcal{L}^{N-}$, and $\mathcal{L}^N$ via boundedness and vanishing of imaginary parts of remainders.
• To demonstrate that the hierarchy of Löwner classes does not collapse in two variables, unlike in one variable, by constructing a counterexample.

Proposed method

• Uses the two-variable Nevanlinna representation to express Pick functions as operator inner products, enabling moment-theoretic analysis.
• Applies Agler-McCarthy vector moment theory to translate asymptotic regularity into operator-theoretic moment conditions.
• Employs spectral theory and monotone convergence to analyze the asymptotic behavior of imaginary parts of functions along non-tangential paths at infinity.
• Derives a recursive structure for residues via the $r_k(z)$ functions, which are inner products of resolvent-related vectors.
• Introduces the concept of directional residues and scalar/vector moments to characterize regularity up to order $2N+1$, linking them to polynomial derivatives.
• Constructs a counterexample in $l^2(\mathbb{Z}_{2(n-1)})$ using a modified left regular representation to show $\mathcal{L}^N \neq \mathcal{L}^{N-}$.

Experimental results

Research questions

• RQ1 Under what conditions does a two-variable Pick function with real residues to order $2N-1$ admit real residues to order $2N$ and directional residues to order $2N+1$?
• RQ2 When is a function in the intermediate Löwner class $\mathcal{L}^{N-}$ also in the full Löwner class $\mathcal{L}^N$?
• RQ3 What is the precise relationship between the existence of scalar and vector moments and the asymptotic expansion of the imaginary part of a Pick function at infinity?
• RQ4 How does the two-variable theory differ from the one-variable case, particularly in the collapse of the Löwner class hierarchy?
• RQ5 What characterizes the $2N+1$-th derivative of a two-variable Pick function in terms of moment conditions and residue structure?

Key findings

• A two-variable Pick function $f$ has real residues to order $2N+1$ if and only if its $2N+1$-th derivative is a polynomial, providing a complete characterization in the spirit of the classical Julia-Carathéodory theorem.
• The function $f$ lies in $\mathcal{L}^N$ if and only if it lies in $\mathcal{L}^{N-}$ and the remainder after subtracting the $2N-1$-order expansion has imaginary part vanishing to order $2N+1$, ensuring higher-order regularity.
• The hierarchy $\mathcal{L}^N \neq \mathcal{L}^{N-}$ holds in two variables, as demonstrated by a concrete counterexample where $r_{2n-1}(z)$ is not a polynomial despite lower-order $r_k(z)$ being polynomials.
• The imaginary part of the remainder $s^{2N-1} \text{Im}[h(isb) - \sum_{|n|\leq 2N-3} \rho_n/(isb)^n]$ is bounded for all $b \in (\mathbb{R}^+)^2$ if and only if $h \in \mathcal{L}^{N-}$, establishing a key inductive criterion.
• The directional residue $r_{2N-1}(b)$ is equal to $-\sum_{|n|=2N-1} \rho_n / b^n$, showing a precise algebraic relation between residues and the leading-order remainder.
• The counterexample function $f(z) = \langle (A - z_Y)^{-1} \alpha, \alpha \rangle$ lies in $\mathcal{L}^{N-}$ but not in $\mathcal{L}^N$, since $r_{2n-1}(z)$ is not a polynomial, proving the strict inclusion $\mathcal{L}^N \subsetneq \mathcal{L}^{N-}$.