Fractal dimensions of spectral measures of rank one perturbations of a positive self-adjoint operator

Abstract

Let H be a Hilbert space. Suppose A is a positive self-adjoint operator on H and φ∈H is a cyclic unit vector. For each λ∈R, we can define the rank one perturbation of A by Aλ=A+λ〈φ,⋅〉φ. To each Aλ we can consider the spectral measure of φ, which we denote by μλ. This generates a family of measures, {μλ}, and we analyze the packing dimension of this family. Past results have determined that the Hausdorff dimension of this family can be determined if the limit inferior of a ratio involving μ is constant on a Lebesgue typical set. This ratio is sometimes called the pointwise dimension of μ and is related to the upper derivative of μ. Work has been done to make a similar argument for the packing dimension, but with little success. Using the theory of rank one perturbations and Borel transforms, we introduce the concept of Lebesgue exact dimension for μ, which allows us to determine the packing dimension of spectral measures of almost every rank one perturbation μλ. If the Lebesgue exact dimension for μ is 1<α<2 then the packing dimension of Lebesgue almost every μλ is 2−α. As a corollary, we find that this limit condition implies a stronger result: the Hausdorff and packing dimensions are equal for almost every μλ.