Characterization of $k-$smooth operators between Banach spaces

TL;DR

This paper investigates the concept of $k$-smoothness for bounded linear operators between Banach spaces, providing characterizations for operators from finite-dimensional $ ext{l}_1^n$ spaces and between two-dimensional Banach spaces.

Contribution

It offers new characterizations of $k$-smooth operators in finite and two-dimensional Banach spaces, extending the understanding of operator smoothness.

Findings

Characterization of $k$-smooth operators from $ ext{l}_1^n$ to Banach spaces

Complete characterization of $k$-smooth operators between two-dimensional Banach spaces

Extension of $k$-smoothness concepts to arbitrary Banach spaces

Abstract

We study $k-$smoothness of bounded linear operators defined between arbitrary Banach spaces. As an application, we characterize $k-$smooth operators defined from $\ell_1^n$ to an arbitrary Banach space. We also completely characterize $k-$smooth operators defined between arbitrary two-dimensional Banach spaces.