Let M be a separable properly infinite semifinite factor. We show that every normal operator T in M is an arbitrarily small max{||·||, ||·||_p}-norm perturbation of a diagonal operator whenever p>dimH(σ_e(T)), where σ_e(T) is the essential spectrum of T and dimH(F) denotes the Hausdorff dimension of a compact set F in the complex plane. On the other hand, if dimH(F)>1, then there exists a normal operator T in M with σ_e(T)=F such that T is not a max{||·||,||·||_p}-norm perturbation of a diagonal operator for any p<dimH(F). Furthermore, we establish an analogous result for unitarily invariant norms when replacing the Hausdorff dimension with the lower Minkowski dimension.

马明辉，大连理工大学博士后，研究方向为算子理论与算子代数。