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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 \(n\to\infty\).

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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