Operators Associated with Soft and Hard Spectral Edges from 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 is simply invariant under the translation semigroup e (t ≥ 0) and invariant under the Schrödinger semigroup e +x) (t ≥ 0); these properties characterize WL 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 L2(0,∞) 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.