Blower, Gordon (2008) Operators associated with soft and hard spectral edges from unitary ensembles. Journal of Mathematical Analysis and Applications, 337 (1). pp. 239-265. ISSN 0022-247X
Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of integrable operators associated with soft and hard edges of eigenvalues distributions of random matrices. Such Tracy--Widom operators are realized as controllability operators for linear systems, and are reporducing 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 Tracy--Widom operators for the soft edge of the unitary ensemble and hard edge of the Jacobi ensemble.
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