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 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 L 2 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.

Integrable operators and the squares of Hankel operators

“…The general theorem of [2] specialised to the Airy and Bessel kernels, and in this paper we prove related results which deal with other integrable operators.…”

“…In [2] we considered a general class of differential equations which gives rise to integrable operators that have the form W = Γ * Γ , where Γ : L 2 (0, ∞) → L 2 ((0, ∞); K) is a continuous Hankel operator. The general theorem of [2] specialised to the Airy and Bessel kernels, and in this paper we prove related results which deal with other integrable operators.…”

“…By Lemma 2.1 of [2], which essentially depends upon the continuity of the Hilbert transform on L 2 (R), we know that W is also a continuous linear operator on L 2 (0, ∞).…”

“…. , m. In[2] we used Loewner's characterization of operator monotone functions, which shows in particular that an operator monotone function on (0, ∞) extends to an analytic function on a domain U containing (0, ∞) as in[9, p. 541]. Theorem 1.2 shows that, under mild technical conditions, W admits of a factorization…”

Hankel operators that commute with second-order differential operators

“…In [2,3] we considered several differential equations of the form (1.5) and resolved whether factorization of W takes place, giving explicit formulae for φ; several important examples involve the confluent hypergeometric equation, which may be reduced by change of variable to Whittaker's equation. For a more general statement about reducing systems, and how changes of variable can affect factorization, see [3,Section 3].…”