International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 611-614
doi:10.1155/S0161171291000832
Research notes
A note on best approximation and invertibility of operators on uniformly convex Banach spaces
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061, Virginia, USA
Received 1 October 1990
Copyright © 1991 James R. Holub. 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.
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−12S‖<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.