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Transactions of the American Mathematical Society

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

 

 

Spectral mapping theorems and perturbation theorems for Browder's essential spectrum


Author: Roger D. Nussbaum
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
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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,\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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DOI: https://doi.org/10.1090/S0002-9947-1970-0265967-9
Keywords: Browder's essential spectrum, spectral mapping theorems, perturbation theorems, noncompact perturbations
Article copyright: © Copyright 1970 American Mathematical Society