Tingley's problem for $p$-Schatten von Neumann classes

Abstract

Let $H$ and $H'$ be the complex Hilbert spaces. For $p\in]1,\infty\[\backslash{2}$ we consider the Banach space $C\_p(H)$ of all $p$-Schatten von Neumann operators, whose unit sphere is denoted by $S(C\_p(H))$. In this paper we prove that every surjective isometry $\Delta\colon S(C\_p(H))\to S(C\_p(H'))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T\colon C\_p(H)\to C\_p(H')$.