A complete characterization of smoothness in the space of bounded linear operators

TL;DR

This paper provides a comprehensive characterization of smoothness for bounded linear operators in infinite-dimensional normed spaces using Birkhoff-James orthogonality and semi-inner-products, highlighting the role of norming sequences and the norm attainment set.

Contribution

It introduces a novel characterization of operator smoothness based on norming sequences and explores the equivalence of Gâteaux and Fréchet differentiability for compact operators in specific Banach spaces.

Findings

Complete characterization of smoothness using norming sequences.

Importance of the norm attainment set in smoothness analysis.

Equivalence of Gâteaux and Fréchet differentiability for certain compact operators.

Abstract

We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that G\^{a}teaux differentiability and Fr\'{e}chet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fr\'{e}chet differentiable normed linear space, endowed with the usual operator norm.