Operators associated with the soft and hard spectral edges of unitary ensembles

Abstract

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the operators associated with soft and hard edges of eigenvalue distributions of random matrices. Tracy and Widom introduced a projection operator $W$ to describe the soft edge of the spectrum of the Gaussian unitary ensemble. The subspace $WL^2$ is simply invariant under the translation semigroup $e^{itD}$ $(t\geq 0)$ and invariant under the Schr\"odinger semigroup $e^{it(D^2+x)}$ $(t\geq 0)$; these properties characterize $WL^2$ via Beurling's theorem. The Jacobi ensemble of random matrices has positive eigenvalues which tend to accumulate near to the hard edge at zero. This paper identifies a pair of unitary groups that satisfy the von Neumann--Weyl anti-commutation relations and leave invariant certain subspaces of $L^2(0,\infty)$ which are invariant for operators with Jacobi kernels. Such Tracy--Widom operators are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy--Widom type operators.