Generalized \(a\)-Weyl’s theorem and the single-valued extension property. (English) Zbl 1123.47005

The author investigates the generalized \(a\)-Weyl theorem and the generalized \(a\)-Browder theorem for Banach space operators \(T\) for which \(T\) or \(T^*\) has the single-valued extension property. He proves that the spectral mapping theorem holds for the semi-essential approximate point spectrum of such operators. Moreover, he gives some applications for bounded linear operators \(T\) on a complex Banach space \(X\) for which there is a function \(p:\mathbb{C}\to\mathbb{Z}^+\) such that \[ H_0(T-\lambda):=\{x\in X:\lim_{n\to+\infty}\| (T-\lambda)^nx\| ^{\frac{1}{n}}=0\}=\ker(T-\lambda)^{p(\lambda)} \] for all \(\lambda\in\mathbb{C}\).


47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A20 Dilations, extensions, compressions of linear operators
47A53 (Semi-) Fredholm operators; index theories
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