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

TL;DR

This paper proves that every surjective isometry on the unit sphere of p-Schatten von Neumann classes extends to a linear or conjugate linear isometry on the entire space, for p in (1, ∞) excluding 2.

Contribution

It establishes the extension property of isometries on the unit sphere of p-Schatten classes for p in (1, ∞) ackslash {2}, a problem known as Tingley's problem.

Findings

Surjective isometries on the sphere extend to the whole space.

Extension holds for all p in (1, ∞) ackslash {2}.

Results contribute to the understanding of geometric structure of Schatten classes.

Abstract

Let $H$ and $H'$ 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 $\Delta: S(C_p(H))\to S(C_p(H'))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T:C_p(H)\to C_p(H')$.