International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 257-261

Nontrivial isometries on sp(α)

Stephen L. Campbell

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.


sp(α) is a Banach space of sequences x with x=(i=0|xi|p+αi=0|xi+1xi|p)1/p. For 1<p<, p2, 0<α<, α1, 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.