On a Generalized Weyl-Von Neumann Converse Theorem
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.
KeywordsCompact Operator Banach Algebra Essential Spectrum Weyl Algebra Proper Ideal
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