2006/06/10 by Vadim Kostrykin, Kostrykin, Vadim, Konstantin A. Makarov +1
Mathematics · #45P05 #47A55 #47B35 #FOS: Mathematics #Holomorphic and Operator Theory #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math/0606249
openalex publication_date 2006/06/10 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
In his 1953 paper [Matem.~Sbornik 33 (1953), 597 -- 626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint 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.