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Browder’s theorems and spectral continuity. (English) Zbl 0979.47004

A bounded operator \(T\) on some complex Banach space is said to obey Browder’s theorem if its Weyl spectrum equals \(\sigma(T)\setminus \pi_0(T)\) with \(\pi_0(T)\) being the set of all normal eigenvalues of \(T\). In this article, the author introduces operators that obey the \(a\)-Browder theorem, a property which turns out to be more restrictive than obeying the usual Browder theorem. On the other hand, by characterizing the operators obeying \(a\)-Browder’s theorem in terms of other spectral data it is shown that quasinilpotent operators, algebraic operators and polynomially Riesz operators all obey the \(a\)-Browder theorem. Moreover, it is proved that if \(a\)-Browder’s theorem holds for \(T\), and \(p\) is any polynomial, then \(p(T)\) obeys \(a\)-Browder’s theorem if and only if \(p(\sigma_{ea}(T))= \sigma_{ea}(p(T))\), where \(\sigma_{ea}\) denotes the essential approximative point spectrum.

MSC:

47A10 Spectrum, resolvent
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