The Tingley problem on the unit sphere of complex Lp[0,1] space

Abstract

In this paper, we investigate the problem of extending isometric operators from unit sphere of complex Lp spaces (1<p<∞, p≠2) to general complex Banach spaces. By studying the isometric operators, we prove the Tingley problem on complex Lp spaces and provide a positive answer under some conditions. That is, it is proved that for a surjective isometry V₀ on any complex Lp[0,1] unit sphere to any general complex Banach space E unit sphere, Under some conditions,V₀ can be extended to a linear isometry from the entire space Lp[0,1] to E.