Weyl’s and Browder’s theorems for operators satisfying the SVEP.

*(English)*Zbl 1064.47004A 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.

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.

Reviewer: Manuel GonzĂˇlez (Santander)