The Leray-Schauder alternative for nonexpansive maps from the ball characterize Hilbert space (Q2753309)

A nonexpansive operator \(T\) from the unit ball \(B(X)\) of a real Banach space \(X\) into \(X\) satisfies the Leray-Schauder alternative if either \(T\) has a fixed point in \(B(X)\), or \(T\) has an eigenvalue \(\lambda> 1\) with eigenvector of norm \(1\). In this interesting note the authors show that, loosely speaking, these two alternatives exclude each other if and only if \(X\) is a Hilbert space.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].