Extending Weyl-von Neumann Theorem on perturbation of self-adjoint operators, in 1979 Voiculescu was able to show that a normal operator in B(H) is a diagonal operator plus an arbitrary small Hilbert-Schmitz operator where H is a separable Hilbert space. On the other hand, the classical Kato-Rosenblum theorem states that the absolute continuous part of a self-adjoint operator in B(H) can't be changed, relative to unitary equivalence, by a perturbation of a trace class operator.
Assume that M is a properly infinite separable von Neumann algebra with a faithful normal semifinite tracial weight (an easy example of such M is B(H) with a canonical trace). In the first half of the talk, we will prove that a normal operator in a separable seminifinite von Neumann algebra is a diagonal operator plus an arbitrary small max(\|.\|, \|.\|_2)-perturbation in M. We will devote our second half of the talk to show that, if H is the real part of a non-normal hyponormal operator in M, then H can't be a diagonal operator plus a trace class operator in M.
沈隽皓，美国新罕布什尔大学数学与统计学院终身教授。沈隽皓教授于2004年毕业于美国宾夕法尼亚大学，师从著名算子理论与算子代数专家Richard Kadison。毕业后任职于美国新罕布什尔大学，长期从事泛函分析，算子理论与算子代数的研究工作，研究成果卓著, 在《Journal of Functional Analysis》，《Transactions of the American Mathematical Society》，《Journal of Operator Theory》，《Canadian Journal of Mathematics》，《The Bulletin of the London Mathematical Society》，《Integral Equations and Operator Theory》等数学名刊上发表论文30余篇。