A note on best approximation and invertibility of operators on uniformly convex Banach spaces - James R. Holub (International Journal of Mathematics and Mathematical Sciences)
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.