On k-smoothness of operators between Banach spaces

TL;DR

This paper investigates the concept of k-smoothness of bounded linear operators between Banach spaces, introducing a new index of smoothness, and characterizes this property for operators between Hilbert and polyhedral Banach spaces.

Contribution

It introduces the index of smoothness for operators, providing new characterizations of k-smoothness between Hilbert and polyhedral Banach spaces, extending existing results.

Findings

Characterization of k-smoothness for operators between Hilbert spaces.

Dependence of rank 1 operator smoothness on space dimension.

Generalization and improvement of previous k-smoothness results.

Abstract

We explore the $k$-smoothness of bounded linear operators between Banach spaces, using the newly introduced notion of index of smoothness. The characterization of the $k$-smoothness of operators between Hilbert spaces follows as a direct consequence of our study. We also investigate the $k$-smoothness of operators between polyhedral Banach spaces. In particular, we show that the $k$-smoothness of rank $1$ operators between polyhedral spaces depends heavily on the dimension of the corresponding spaces rather than the geometry of the spaces. The results obtained in this article generalize and improve upon the existing results in the $k$-smoothness of operators between Banach spaces.