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A note on iterates that are contractions. (English) Zbl 0602.47002
In this note it is proved: Theorem: Let T:B$$\to B$$ be a continuous linear selfmap of a B-space. Then $$\| T^ N\| <1$$ for some N if and only if $$\sum \| T^ n(x)\| <\infty$$ for all x in B.

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
contraction
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##### References:
 [1] Browder, F.E; Petryshyn, W.V, The solution by iteration of linear functional equations in Banach spaces, Bull. amer. math. soc., 72, 566-570, (1966) · Zbl 0138.08201 [2] Rudin, W, Real and complex analysis, (1966), McGraw-Hill New York · Zbl 0148.02904 [3] Schechter, M, Principles of functional analysis, (1971), Academic Press New York · Zbl 0211.14501
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