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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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##### References:
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