A note on best approximation and invertibility of operators on uniformly convex Banach spaces (Q1178805)

It is well-known that if \(S\) is a bounded linear operator on a Banach space \(X\) for which \(\| I-S\|<1\), then \(S\) is invertible. Equivalently, if \([S]\) denote the subspace of \({\mathcal L}(X)\) spanned by \(S\), then \(S\) is invertible if \(dist(I,[S])<1\). In this paper the author considers bounded linear operators \(S\) on a uniformly convex Banach space \(X\) for which \(\| I-S\|=1\). For such operators the author shows that \(S\) is invertible if and only if \(\| I-{1\over2}S\|<1\). 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 \(dist(I,[S])<1\).