The Weak differentiability of norm and a generalized Bhatia-\v{S}emrl Theorem

TL;DR

This paper characterizes weak differentiability of norms in multilinear map spaces and generalizes the Bhatia-emrl theorem to multilinear contexts, advancing understanding of norm behavior and orthogonality.

Contribution

It provides a complete characterization of weak differentiability of norms in multilinear spaces and extends the Bhatia-emrl theorem to multilinear maps.

Findings

Complete characterization of weak differentiability in multilinear spaces

Multilinear generalization of Bhatia-emrl theorem

Enhanced understanding of orthogonality in multilinear contexts

Abstract

We completely characterize the weak differentiability (or, in other words Gateaux differentiability) of the norm in the spaces of bounded multilinear maps. Also, we obtain a multilinear generalization of the well-known Bhatia-\v{S}emrl theorem on Birkhoff-James orthogonality.