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On a Generalized Weyl-Von Neumann Converse Theorem

  • M. Seddighin
  • K. Gustafson

Abstract

For two bounded selfadjoint operators A and A + B with the same essential spectra, there exists a unitary operator U such that B − (UAU* − A) is compact (Weyl-Von Neumann). More generally, an operator B in B(H) is compact iff σe (A + B) = σe (A) for all A ∈ B(H) (Gustafson-Weidmann), and in fact one needs only σ(A + B) ∩ σ (A) not empty for all A ∈ B(H) (Dyer, Porcelli, Rosenfeld). Aiken (Is. Math. J., 1976) and Zemanek (Studia Math., to appear) have studied the question of when for an arbitrary Banach algebra with identity the last condition guarantees that B is in some proper two-sided ideal. We give new results for this question, including a number of examples.

Keywords

Compact Operator Banach Algebra Essential Spectrum Weyl Algebra Proper Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • M. Seddighin
    • 1
    • 2
  • K. Gustafson
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Mashad UniversityUSA

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