Integrable operators and squares of Hankel Matrices

TL;DR

This paper establishes conditions under which Tracy-Widom type operators, characterized by a specific kernel form, can be expressed as squares of Hankel operators, with applications to random matrix theory.

Contribution

It provides new sufficient conditions for Tracy-Widom type operators to be squares of Hankel operators, extending understanding in both discrete and integral operator contexts.

Findings

Applicable to the discrete Bessel kernel in random matrix theory

Provides criteria for operator factorization as Hankel squares

Bridges discrete and continuous operator frameworks

Abstract

In this note, we find sufficient conditions for an operator with kernel of the form $A(x)B(y)-A(x)B(y)/(x-y)$ (which we call a Tracy-Widom type operator) to be the square of a Hankel operator. We consider two contexts: infinite matrices on $\ell^2$, and integral operators on the Hardy space $H^2(\mathbb{T})$. The results can be applied to the discrete Bessel kernel, which is significant in random matrix theory.