Semigroups of mappings with rigid Lipschitz constant (Q2781299)

Let \(B_2^+\) denote the positive part of the unit ball in \(\ell_2\). \textit{T. Kuczumow} [Proc. Am. Math. Soc. 127, No. 9, 2671-2678 (1999; Zbl 0921.47049)] asked whether or not there exists a renorming of \(\ell_2\) which makes the Baillon mapping nonexpansive on \(B_2^+\). The author shows that, for any equivalent norm on \(\ell_2\), and for each \(0 < k < \pi/2\), the Baillon mapping is not uniformly Lipschitzian on \(B_2^+\). Therefore it cannot be nonexpansive on \(B_2^+\) with respect to any equivalent renorming of \(\ell_2\). He also demonstrates that the same is true for the Goebel-Kirk-Thele and Kakutani mappings.