International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1991 / Article

Research notes | Open Access

Volume 14 |Article ID 567578 | https://doi.org/10.1155/S0161171291000832

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, Article ID 567578, 4 pages, 1991. https://doi.org/10.1155/S0161171291000832

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

Received01 Oct 1990

Abstract

It is shown that if X is a uniformly convex Banach space and S a bounded linear operator on X for which IS=1, then S is invertible if and only if I12S<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.

Copyright © 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.


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