The Weyl-von Neumann theorem in von Neumann factors

In a von Neumann factor M of type II∞ with semifinite faithful normal trace τ, the q-th Schattentype norm ‖·‖q of an operator A ϵ M is defined as ‖A‖q = (τ(|A|q))1/q.We will prove that for every self-adjoint operator H ϵ M and any numbers q > 1, ε > 0, there exists a self-adjoint perturbation operator A ϵ M such that ‖A‖ < ε, ‖A‖q < ε and H + A has a pure point spectrum.We will also discuss the possibility of transferring this result onto other von Neumann factors.

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Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).