Diagonalization modulo norm ideals and Hausdorff dimensionality

摘要

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal C not contained in the trace class C-1, every self-adjoint operator A admits the decomposition A = D + K, where D is a self-adjoint diagonal operator and K is an element of C. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if mu is a regular Borel measure on X which has a sigma -finite one-dimensional Hausdorff measure, then the family (M-f: f is an element of Lip(X)} of multiplication operators on L-2(X, mu) can be simultaneously diagonalized module any C not equal C-1. Because the condition C not equal C-1 in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between C-1 and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p > 1. [References: 19]