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arXiv · 2402.00574

B-Fredholm theory in Banach algebras

Yunnan Zhang·Qingping Zeng·Zhenying Wu
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Abstract

The aim of this paper is to develop a systematic B-Fredholm theory in semiprime Banach algebras. We first generalize Smyth's important punctured neighbourhood theorem to B-Fredholm elements. Then using this result, we investigate the local spectral theory of B-Fredholm elements, including the localized left (resp. right) SVEP and a classification of components of B-Fredholm resolvent set. Finally, in semisimple Banach algebra context, we characterize element $f$ such that $f^{n}$ belongs to the socle for some $n \in \mathbb{N}$ from two different perspectives: one is the invariance of the B-Fredholm spectrum under commuting perturbation $f$, the other is the Rieszness and the B-Fredholmness of $f$.

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Yunnan Zhang, Qingping Zeng, Zhenying Wu. 2024-02-01. B-Fredholm theory in Banach algebras. https://arxiv.org/abs/2402.00574

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