Operators associated with soft and hard spectral edges from unitary ensembles (Q2382699)

The motivation for this paper arises on operators that are associated with random-matrix ensembles. More concretely, \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); ibid. 163, No. 1, 33--72 (1994; Zbl 0813.35110)] used several operators to describe the soft edge of the spectrum of the Gaussian unitary ensemble. In the present paper, the author considers similar kernels of the form \[ W(x,y)=\frac{A(x)B(y)-A(y)B(x)}{x-y}, \] where \(A,B\) are square-integrable continuous real functions vanishing at infinity. The main theorem of the paper gives a sufficient condition for the integral operator of such a kernel \(W\) to be the square of a selfadjoint Hankel integral operator. In further sections, the author explores the relation of these kernels with Marchenko's integral equation, reproducing kernels for weighted Hardy spaces, soft-edge operators, hard-edge operators, Sonine spaces, and Mathieu's equation.