# On infinite Jacobi matrices with a trace class resolvent

## TL;DR

This paper investigates conditions under which the resolvent of an infinite Jacobi matrix with a trace class inverse has a convergent series representation, linking spectral properties to orthogonal polynomials and providing specific examples.

## Contribution

It establishes a new criterion for the convergence of a series related to Jacobi matrices with trace class inverse and applies it to Al-Salam-Carlitz II polynomials.

## Key findings

- Series converges locally uniformly on  under given conditions.
- The spectral measure is characterized by an infinite product.
- Application to Al-Salam-Carlitz II polynomials demonstrates the theorem.

## Abstract

Let $\{\hat{P}_{n}(x)\}$ be an orthonormal polynomial sequence and denote by $\{w_{n}(x)\}$ the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for $\{\hat{P}_{n}(x)\}$ is determinate and denote by $J$ the corresponding Jacobi matrix operator on $\ell^{2}$. We show that if $J$ is positive definite and $J^{-1}$ belongs to the trace class then the series on the right-hand side of the defining equation \[ \mathfrak{F}(z):=1-z\sum_{n=0}^{\infty}w_{n}(0)\hat{P}_{n}(z) \] converges locally uniformly on $\mathbb{C}$ and it holds true that $\mathfrak{F}(z)=\prod_{n=1}^{\infty}(1-z/\lambda_{n})$ where $\{\lambda_{n};\,n=1,2,3,\ldots\}=\mathrm{Spec}\,J$. Furthermore, the Al-Salam-Carlitz II polynomials are treated as an example of orthogonal polynomials to which this theorem can be applied.