On Krein’s Example

In his 1953 paper [Matem. Sbornik 33 (1953), 597 – 626] Mark Krein presented an example of a symmetric rank one perturbation of a selfadjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval [−1, 1] and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm. The spectral shift function plays a very important role in perturbation theory for self-adjoint operators. It was introduced in a special case by I. Lifshitz [8] and in the general case (in the framework of trace class perturbations) by M. Krein in his celebrated 1953 paper [7]. He showed that for a pair of self-adjoint not necessarily bounded operators A0 and A1 such that their difference A1−A0 is trace class there exists a unique function ξ ∈ L1(R) satisfying the trace formula (1) tr(φ(A1)− φ(A0)) = ∫