Rank one perturbations and singular integral operators

Rank one perturbations of self-adjoint operators

Operators with Singular Continuous Spectrum, Iv. Hausdorff Dimensions, Rank One Perturbations, and Localization

Although concrete operators with singular continuous spectrum have proliferated recently [7,11,13,17,34,35,37,39], we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and studied recen...

Operators with Singular Continuous Spectrum, Iv. Hausdorff Dimensions, Rank One Perturbations, and Localization

Operators with Singular Continuous Spectrum: Ii. Rank One Operators

For an operator, A, with cyclic vector φ, we study A+ λP where P is the rank one projection onto multiples of φ. If [α,β] ⊂ spec(A) and A has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. §

ar X iv : 0 81 0 . 27 50 v 1 [ m at h . FA ] 1 5 O ct 2 00 8 RANK ONE PERTURBATIONS AND SINGULAR INTEGRAL OPERATORS

We consider rank one perturbations Aα = A+α( · , φ)φ of a self-adjoint operator A with cyclic vector φ ∈ H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ’rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two w...