International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 257-261
doi:10.1155/S0161171282000222
Nontrivial isometries on sp(α)
Department of Mathematics, North Carolina State University, Raleigh 27650, North Carolina, USA
Received 12 January 1981
Copyright © 1982 Stephen L. Campbell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
sp(α) is a Banach space of sequences x with ‖x‖=(∑i=0∞|xi|p+α∑i=0∞|xi+1−xi|p)1/p. For 1<p<∞, p≠2, 0<α<∞, α≠1, there are no nontrivial surjective isometries in sp(α). It has been conjectured that there are no nontrivial isometries. This note gives two distinct counterexamples to this conjecture and a partial affirmative answer for the case of isometries with finite codimension.