Operators associated with soft and hard spectral edges from unitary ensembles

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy–Widom operators are realized as controllability operators for linear systems, and 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. This paper identifies a pair of unitary groups that satisfy the von Neumann–Weyl anti-commutation relations and leave invariant the subspaces of L2 that are the ranges of projections given by the Tracy–Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble.