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Weyl’s and Browder’s theorems for operators satisfying the SVEP. (English) Zbl 1064.47004
A continuous operator $$T$$ acting on a complex Banach space is a Weyl operator if it is Fredholm of index zero; and $$T$$ is a Browder operator if it is Fredholm of finite ascent and descent. From these classes we define the Weyl spectrum $$\sigma_w(T)$$ and the Browder spectrum $$\sigma_b(T)$$ in the obvious way. We denote by $$\pi_{00}(T)$$, $$\pi_0(T)$$ the sets of isolated points $$\lambda$$ of $$\sigma(T)$$ such that $$0<\dim N(T-\lambda) <\infty$$, $$\lambda$$ is a Riesz point of $$T$$, respectively. We say that Weyl’s theorem holds for $$T$$ if $$\sigma_w(T) =\sigma(T) \setminus \pi_{00}(T)$$; and we say that Browder’s theorem holds for $$T$$ if $$\sigma_w(T) =\sigma(T) \setminus \pi_0(T)$$.
In most of the results of this paper, $$T$$ is an operator such that $$T$$ or $$T^*$$ has the single-valued extension property (SVEP). In this case, it is shown that the analytic spectral mapping theorem holds for $$\sigma_w(T)$$, several conditions equivalent to Weyl’s theorem holds for $$T$$ are given, and it is shown that Browder’s theorem holds for all $$f(T)$$, $$f\in H(\sigma(T))$$.
These results are applied to study a certain class of operators $$\Sigma(X)$$. It is shown that, if there exists a function $$h\in H(\sigma(T))$$ which is identically constant in no connected component of its domain and satisfies $$h(T)\in\Sigma(X)$$, then Weyl’s theorem holds for $$f(T)$$ and $$f(T^*)$$ for all $$f\in H(\sigma(T))$$. The class $$\Sigma(X)$$ includes the totally paranormal, the subscalar and some other classes of operators. Thus a unified proof is given for some previously known results.

##### MSC:
 47A11 Local spectral properties of linear operators 47A53 (Semi-) Fredholm operators; index theories
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