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

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1991-01-01

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Hindawi Publishing Corp

Abstract

It is shown that if X is a uniformly convex Banach space and S a bounded linear operator onX for which ?I-S?=1, then S is invertible if and only if ?I-12S? <1. From this it follows thatif S is invertible on X then either (i) dist(I,[S])<1, or (ii) 0 is the unique best approximation toI from [S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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Citation

James R. Holub, "A note on best approximation and invertibility of operators on uniformly convex Banach spaces," International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 611-614, 1991. doi:10.1155/S0161171291000832