# Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 101-130.

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

**Authors**:
Akash Anand (1), Sameer Chavan (2), Rajkamal Nailwal (3)

**Author institution:**(1) Department of Mathematics and Statistics,
Indian Institute of Technology Kanpur, Kanpur, 208016, India

(2) Department of Mathematics and Statistics,
Indian Institute of Technology Kanpur, Kanpur, 208016, India

(3) Department of Mathematics and Statistics,
Indian Institute of Technology Kanpur, Kanpur, 208016, India

**Summary: ** We discuss the problem of classifying polynomials $p : \mathbb R^2_+ \rightarrow (0, \infty)$ for which
$\frac{1}{p}=\{\frac{1}{p(m, n)}\}_{m, n \geqslant 0}$ 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 $b, c, d \geqslant 0,$ then
$\frac{1}{p}$ is joint completely monotone if and only if $a d - b c \leqslant 0.$
We also present an application to the Cauchy dual subnormality problem for toral $3$-isometric weighted $2$-shifts.

**DOI: **http://dx.doi.org/10.7900/jot.2022aug14.2426

**Keywords: ** rational function, joint completely monotone, Bessel function, toral $m$-isometry, Cauchy dual

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