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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Spectral mapping theorems and perturbation theorems for Browder’s essential spectrum
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by Roger D. Nussbaum PDF
Trans. Amer. Math. Soc. 150 (1970), 445-455 Request permission

Abstract:

If $T$ is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of $T,\operatorname {ess} (T)$ [1]. We derive below spectral mapping theorems and perturbation theorems for Browder’s essential spectrum. If $T$ is a bounded linear operator and $f$ is a function analytic on a neighborhood of the spectrum of $T$, we prove that $f(\operatorname {ess} (T)) = \operatorname {ess} (f(T))$. If $T$ is a closed, densely defined linear operator with nonempty resolvent set and $f$ 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 $\operatorname {ess} (T + B) \subseteq \operatorname {ess} (T) + \operatorname {ess} (B) = \{ \mu + v:\mu \in \operatorname {ess} (T),v \in \operatorname {ess} (B)\}$. By making additional assumptions, we obtain an analogous theorem for $B$ unbounded.
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Additional Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 150 (1970), 445-455
  • MSC: Primary 47.30
  • DOI: https://doi.org/10.1090/S0002-9947-1970-0265967-9
  • MathSciNet review: 0265967