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Operators associated with soft and hard spectral edges from unitary ensembles.

Research output: Contribution to journalJournal article

Published

Journal publication date1/01/2008
JournalJournal of Mathematical Analysis and Applications
Journal number1
Volume337
Number of pages27
Pages239-265
Original languageEnglish

Abstract

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.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 337 (1), 2008, © ELSEVIER.