Diagonalization modulo norm ideals and Hausdorff dimensionality

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal C not contained in the trace class C₁, every self-adjoint operator A admits the decomposition A = D + K, where D is a self-adjoint diagonal operator and K ∈ C. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {M_f : f ∈ Lip(X)} of multiplication operators on L²(X, μ) can be simultaneously diagonalized modulo any C ≠ C₁. Because the condition C ≠ C₁ in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between C₁ and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p > 1.