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On the convergence of Neumann series for noncompact operators. (English) Zbl 0754.47005
Main result: Let $$X$$ be a Banach space, let $$U,K$$ be bounded linear operators on $$X$$ such that $$K$$ is compact and $$\| U\|<1$$. If $$x\in X$$ then the series $$\sum_{n=0}^ \infty(U+K)^ n x$$ converges if and only if $$(U+K)^ n x\to 0$$ as $$n\to\infty$$.

##### MSC:
 47A10 Spectrum, resolvent 31A10 Integral representations, integral operators, integral equations methods in two dimensions 47A50 Equations and inequalities involving linear operators, with vector unknowns
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##### References:
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