Holub, James R.2019-05-202019-05-201991-01-01James 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/S0161171291000832http://hdl.handle.net/10919/89574It 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.application/pdfenCreative Commons Attribution 4.0 InternationalArticle - Refereed2019-05-20Copyright © 1991 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.https://doi.org/10.1155/S0161171291000832