A note on best approximation and invertibility of operators on uniformly convex Banach spaces

James R. Holub

International Journal of Mathematics and Mathematical Sciences, 1991, vol. 14, 1-4

Abstract:

It is shown that if X is a uniformly convex Banach space and S a bounded linear operator on X for which ‖ I − S ‖ = 1 , then S is invertible if and only if ‖ I − 1 2 S ‖ < 1 . From this it follows that if S is invertible on X then either (i) dist ( I , [ S ] ) < 1 , or (ii) 0 is the unique best approximation to I from [ S ] , a natural (partial) converse to the well-known sufficient condition for invertibility that dist ( I , [ S ] ) < 1 .

Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:567578

DOI: 10.1155/S0161171291000832

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