Weyl type theorems for bounded linear operators.

*(English)*Zbl 1050.47014The aim of the paper under review is to show that, from the point of view of Weyl type theorems, the notion of B-Weyl spectrum (see M. Berkani [Integral Equations Oper. Theory 34, 244-249 (1999; Zbl 0939.47010)]) generalizes the notion of Weyl spectrum, as in the case of normal operators on Hilbert spaces. To explain this, we need some notation.

Let \(T\) be a bounded operator on a Banach space. One defines the following sets: the Weyl spectrum \(\sigma_W(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not a Fredholm operator of index \(0\), the B-Weyl spectrum \(\sigma_{BW}(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not a B-Weyl operator of index \(0\), \(\sigma_{{SF_{+}^{-}}}(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not an upper semi-Fredholm operator of negative index, and \(\sigma_{{SBF_{+}^{-}}}(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not an upper semi-B-Fredholm operator of negative index. We also need the following important sets of eigenvalues: \(E_0(T)\) is the set of all eigenvalues of \(T\) of finite multiplicity isolated in the spectrum of \(T\), \(E(T)\) is the set of all eigenvalues of \(T\) isolated in the spectrum of \(T\), and similarly, \(E_0^a(T)\) is the set of all eigenvalues of \(T\) of finite multiplicity isolated in the approximate point spectrum of \(T\), and \(E_0^a(T)\) is the set of all eigenvalues of \(T\) isolated in the approximate point spectrum of \(T\).

Then one says that \(T\) satisfies the generalized \(a\)-Weyl’s theorem, the \(a\)-Weyl’s theorem, the generalized Weyl’s theorem, or Weyl’s theorem, if \(\sigma_{{SBF_{+}^{-}}}(T)=\sigma^a (T)\setminus E^a(T)\), \(\sigma_{{SF_{+}^{-}}}(T)=\sigma^a (T)\setminus E^a_0(T)\), \(\sigma_{BW}(T)=\sigma (T)\setminus E(T)\), or \(\sigma_{W}(T)=\sigma (T)\setminus E_0(T)\), respectively.

The author proves that if \(T\) satisfies the generalized \(a\)-Weyl’s theorem, then \(T\) satisfies the \(a\)-Weyl’s theorem and the generalized Weyl’s theorem. Also, \(T\) satisfies Weyl’s theorem provided that \(T\) obeys the generalized Weyl’s theorem.

Similar results are proved with respect to Browder’s theorem.

Let \(T\) be a bounded operator on a Banach space. One defines the following sets: the Weyl spectrum \(\sigma_W(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not a Fredholm operator of index \(0\), the B-Weyl spectrum \(\sigma_{BW}(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not a B-Weyl operator of index \(0\), \(\sigma_{{SF_{+}^{-}}}(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not an upper semi-Fredholm operator of negative index, and \(\sigma_{{SBF_{+}^{-}}}(T)\) is the set of \(\lambda\in {\mathbb C}\) such that \(T-\lambda I\) is not an upper semi-B-Fredholm operator of negative index. We also need the following important sets of eigenvalues: \(E_0(T)\) is the set of all eigenvalues of \(T\) of finite multiplicity isolated in the spectrum of \(T\), \(E(T)\) is the set of all eigenvalues of \(T\) isolated in the spectrum of \(T\), and similarly, \(E_0^a(T)\) is the set of all eigenvalues of \(T\) of finite multiplicity isolated in the approximate point spectrum of \(T\), and \(E_0^a(T)\) is the set of all eigenvalues of \(T\) isolated in the approximate point spectrum of \(T\).

Then one says that \(T\) satisfies the generalized \(a\)-Weyl’s theorem, the \(a\)-Weyl’s theorem, the generalized Weyl’s theorem, or Weyl’s theorem, if \(\sigma_{{SBF_{+}^{-}}}(T)=\sigma^a (T)\setminus E^a(T)\), \(\sigma_{{SF_{+}^{-}}}(T)=\sigma^a (T)\setminus E^a_0(T)\), \(\sigma_{BW}(T)=\sigma (T)\setminus E(T)\), or \(\sigma_{W}(T)=\sigma (T)\setminus E_0(T)\), respectively.

The author proves that if \(T\) satisfies the generalized \(a\)-Weyl’s theorem, then \(T\) satisfies the \(a\)-Weyl’s theorem and the generalized Weyl’s theorem. Also, \(T\) satisfies Weyl’s theorem provided that \(T\) obeys the generalized Weyl’s theorem.

Similar results are proved with respect to Browder’s theorem.

Reviewer: Ingrid Beltiţă (Bucureşti)