Integrable operators and the squares of Hankel operators (Q2470489)

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large selfadjoint random matrices from the generalized unitary ensembles. \textit{C. A. Tracy} and \textit{H. Widom} [Commun. Math. Phys. 159, No.\,1, 151--174 (1994; Zbl 0789.35152); ibid. 161, No. 2, 289--309 (1994; Zbl 0808.35145)] considered the Airy and Bessel kernels which describe the soft and hard edges of generalized unitary ensembles and proved the apparently miraculous identities that the operators with these kernels were squares of selfadjoint Hankel operators. In the present paper, the author gives sufficient conditions for an integrable operator to be the square of a Hankel operator (Theorem~1.2) and applies the condition to the Airy, associated Laguerre, modified Bessel and Whittaker functions.