On a problem of Nirenberg concerning expanding maps (Q1063853)

Let X be a Banach space. This paper is concerned with the following problem, which was stated by L. Nirenberg: Problem: Let \(T: X\to X\) be a continuous map, which is expanding (\(| Tu-Tv| \geq | u-v|)\) and whose range contains an open set. Does T map X onto X? A negative answer is given, by constructing a simple counterexample in \(L^ 1({\mathbb{N}})\). However, the question remains open as X is a Hilbert space.