Spectral mapping theorems and perturbation theorems for browder’s essential spectrum

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Abstract

If T is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of T, ess (T) [1]. We derive below spectral mapping theorems and perturbation theorems for Browder’s essential spectrum. If T is a bounded linear operator and/is a function analytic on a neighbor­hood of the spectrum of T, we prove that f(ess (T)) = ess (f(T)). If T is a closed, densely defined linear operator with nonempty resolvent set and /is a polynomial, the same theorem holds. For a closed, densely defined linear operator T and a bounded linear operator B which commutes with T, we prove that ess (T+B) Íess (T) + ess (B) = {m, + v: M Î ess (T), v Î ess (B)}. By making additional assumptions, we obtain an analogous theorem for B unbounded.

Original languageEnglish (US)
Pages (from-to)445-455
Number of pages11
JournalTransactions of the American Mathematical Society
Volume150
Issue number2
DOIs
StatePublished - Aug 1970

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Browder’s essential spectrum
  • Noncompact perturbations
  • Perturbation theorems
  • Spectral mapping theorems

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