Joint complete monotonicity of rational functions in two variables and toral $m$-isometric pairs

TL;DR

This paper characterizes when certain rational functions in two variables are jointly completely monotone and applies these results to problems involving toral m-isometric pairs and subnormality of weighted shifts.

Contribution

It provides a complete characterization of joint complete monotonicity for a class of rational functions and connects this to operator theory problems involving toral m-isometric pairs.

Findings

Joint complete monotonicity holds iff a d - b c ≤ 0 for the given polynomial.

Characterization applies to polynomials linear in y with positive a and nonnegative b, c, d.

Application to Cauchy dual subnormality problem for toral 3-isometric weighted 2-shifts.

Abstract

We discuss the problem of classifying polynomials $p : \mathbb R^2_+ \rightarrow (0, \infty)$ for which $\frac{1}{p}=\{\frac{1}{p(m, n)}\}_{m, n \geq 0}$ is joint completely monotone, where $p$ is a linear polynomial in $y.$ We show that if $p(x, y)=a+b x+c y+d xy$ with $a > 0$ and $b, c, d \geq 0,$ then $\frac{1}{p}$ is joint completely monotone if and only if $a d - b c \leq 0.$ We also present an application to the Cauchy dual subnormality problem for toral $3$-isometric weighted $2$-shifts.