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The Weyl-von Neumann theorem in von Neumann factors

Kamiński K.

Journal of Applied Analysis

2017

Vol. 23, nr 1

39--42

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.

A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices

Pchelintseva I.

Opuscula Mathematica

2008

Vol. 28, no. 2

137-150

We consider self-adjoint unbounded Jacobi matrices with diagonal q(n) = b(n)n and off-diagonal entries λ(n) = n, where b(n) is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of b(1)b(2) = 4. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.