### 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 language | English (US) |
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Pages (from-to) | 445-455 |

Number of pages | 11 |

Journal | Transactions of the American Mathematical Society |

Volume | 150 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1970 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

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