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A note on the Browder’s and Weyl’s theorem. (English) Zbl 1163.47005
Let \(T\) be a bounded linear operator on a Banach space \(X\) and let \(\sigma(T)\), \(\sigma_w(T)\), \(\sigma_b(T)\), and \(\sigma_{bw}(T)\) be its spectrum, Weyl spectrum, Browder spectrum, and B-Weyl spectrum, respectively. One says that Weyl’s theorem holds for \(T\) if \(\sigma(T)\setminus\sigma_w(T)=E_0(T)\), where \(E_0(T)\) is the set of all isolated points of \(\sigma(T)\) which are eigenvalues of finite multiplicity. Browder’s theorem holds for \(T\) if \(\sigma_w(T)=\sigma(T)\setminus \pi_0(T)\), where \(\pi_0(T)\) is the set of all isolated points of \(\sigma(T)\) for which the corresponding spectral projection is of finite rank. The generalized Weyl’s theorem says that \(\sigma(T)\setminus\sigma_{bw}(T)=E(T)\), where \(E(T)\) is the set of all isolated points of \(\sigma(T)\) which are eigenvalues. Further, \(T\) is said to satisfy the generalized Browder’s theorem if \(\sigma(T)\setminus\sigma_{bw}(T)=\pi(T)\), where \(\pi(T)\) is the set of all poles of the resolvent of \(T\). Finally, one says that \(T\) has the single-valued extension property (SVEP) at \(\lambda\in{\mathbb C}\) if, for every open disk \(D(\lambda,r)\), the null function is the only analytic solution of the equation \((T-\mu)f(\mu)=0\) for all \(\mu\in D(\lambda,r)\).
The authors show several results relating the above defined concepts. In particular, they prove that the following properties are equivalent:
(i)
\(T\) satisfies Browder’s theorem;
(ii)
\(T\) has the SVEP at \(\lambda\notin\sigma_w(T)\);
(iii)
\(T\) satisfies generalized Browder’s theorem;
(iv)
\(T\) has the SVEP at \(\lambda\notin\sigma_{bw}(T)\).
Another representative result is the following. Suppose that \(E(T)=\pi(T)\). Then the following properties are equivalent:
(a) \(T\) satisfies Weyl’s theorem;
(b) \(T\) satisfies Browder’s theorem;
(c) \(T\) has the SVEP at all \(\lambda\notin\sigma_w(T)\);
(d) \(T\) has the SVEP at all \(\lambda\notin\sigma_{bw}(T)\).
Some applications of the obtained results are given.

MSC:
47A53 (Semi-) Fredholm operators; index theories
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
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