Commutator methods for unitary operators

Commutator methods for unitary operators C. Fernandez 1 , S. Richard 2y and R. Tiedra de Aldecoa 1z 1 Facultad de Matematicas, Ponticia Universidad Catolica de Chile, Av. Vicu~na Mackenna 4860, Santiago, Chile 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan E-mails: , , Abstract We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local niteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the rst time that commutator methods are applied in the context of rotation algebras, for the study of their generators. 2000 Mathematics Subject Classification: 81Q10, 47A35, 47B47. Keywords: Unitary operators, spectral analysis, Mourre theory, cocycles over rotations. 1 Introduction It is commonly accepted that commutator methods (in the sense of E.

• unitary operators induced


• slightly stronger than


• stronger regularity


• adjoint space


• lebesgue spectrum


• divergence-free vector


• operators


• free evolution


• self-adjoint operators