The punctured neighbourhood theorem in Banach algebras. II (Q1977707)

Recall that in a semisimple Banach algebra \(A\) with unity \(e\) and a so-called inessential ideal \(K\), an element \(x\) is \(K\)-left Fredholm if \(yx-e\in K\). It is proved that for such \(x\) and sufficiently small \(|\lambda|>0\) the elements \(x-\lambda e\) are \(K\)-left Fredholm with the same nullity. Moreover, necessary and sufficient conditions are given that this nullity is the nullity of \(x\). The results are proved without using the connection with semi-Fredholm operators. Also generalizations to the case when \(A\) is not semisimple are given. The paper is the continuation of the same author's paper [Proc. R. Ir. Acad. A 91, No. 2, 205-218 (1991; Zbl 0793.46027)].