A perturbation characterization of compactness of self-adjoint operators
Volume 158 / 2003
Studia Mathematica 158 (2003), 199-205 MSC: Primary 47B07, 47A55. DOI: 10.4064/sm158-3-1
A characterization of compactness of a given self-adjoint bounded operator $A$ on a separable infinite-dimensional Hilbert space is established in terms of the spectrum of perturbations. An example is presented to show that without separability, the perturbation condition, which is always necessary, is not sufficient. For non-separable spaces, another condition on the self-adjoint operator $A$, which is necessary and sufficient for the perturbation, is given.