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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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