Frechet differentiability of Sp norms

TL;DR

This paper proves that Schatten-von Neumann norms are Frechet differentiable, connecting this property to recent advances in spectral shift functions, thus resolving a long-standing question in operator theory.

Contribution

It establishes the Frechet differentiability of Schatten-von Neumann norms and links this to recent spectral shift function results, advancing the understanding of operator ideal norms.

Findings

Schatten-von Neumann norms are Frechet differentiable.

Connection established between differentiability and spectral shift functions.

Addresses a long-standing open problem in operator theory.

Abstract

One of the long standing questions in the theory of Schatten-von Neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts. We answer this question in the affirmative. A key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of L.S. Koplienko's conjecture concerning existence of higher order spectral shift functions.