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

Abstract

Let $H$ and $H^{\prime }$ be a 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 $S(C_p(H))$. We prove that every surjective isometry $\Delta :S(C_p(H))\to S(C_p(H^{\prime }))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T:C_p(H)\to C_p(H^{\prime })$.