Weyl spectra and Weyl's theorem

Volume 148 / 2001

Young Min Han, Woo Young Lee Studia Mathematica 148 (2001), 193-206 MSC: Primary 47A10, 47A53, 47A55. DOI: 10.4064/sm148-3-1


“Weyl's theorem” for an operator on a Hilbert space is the statement that the complement in the spectrum of the Weyl spectrum coincides with the isolated eigenvalues of finite multiplicity. In this paper we consider how Weyl's theorem survives for polynomials of operators and under quasinilpotent or compact perturbations. First, we show that if $T$ is reduced by each of its finite-dimensional eigenspaces then the Weyl spectrum obeys the spectral mapping theorem, and further if $T$ is reduction-isoloid then for every polynomial $p$, Weyl's theorem holds for $p(T)$. The results on perturbations are as follows. If $T$ is a “finite-isoloid” operator and if $K$ commutes with $T$ and is either compact or quasinilpotent then Weyl's theorem is transmitted from $T$ to $T+K$. As a noncommutative perturbation theorem, we also show that if the spectrum of $T$ has no holes and at most finitely many isolated points, and if $K$ is a compact operator then Weyl's theorem holds for $T+K$ when it holds for $T$.


  • Young Min HanDepartment of Mathematics
    Sungkyunkwan University
    Suwon 440-746, South Korea
  • Woo Young LeeDepartment of Mathematics
    Sungkyunkwan University
    Suwon 440-746
    South Korea

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