Nonlinear Isometries on Schatten-p Class in Atomic Nest Algebras

Abstract

Let $H$ be a complex Hilbert space; denote by Alg $𝒩$ and ${𝒞}_p(H)$ the atomic nest algebra associated with the atomic nest $𝒩$ on $H$ and the space of Schatten-$p$ class operators on, $H$ respectively. Let ${𝒞}_p(H)\cap \text{Alg\hspace{0.17em}}𝒩$ be the space of Schatten-$p$ class operators in Alg $𝒩$. When and $p\ne 2$, we give a complete characterization of nonlinear surjective isometries on ${𝒞}_p(H)\cap \text{Alg\hspace{0.17em}}𝒩$. If $p=2$, we also prove that a nonlinear surjective isometry on ${𝒞}_2(H)\cap \text{Alg\hspace{0.17em}}𝒩$ is the translation of an orthogonality preserving map.