Characterization of k-smoothness of operators defined between infinite-dimensional spaces

TL;DR

This paper investigates the concept of k-smoothness in bounded linear operators across various infinite-dimensional spaces, providing characterizations and applications to extreme contractions in specific Banach spaces.

Contribution

It offers new characterizations of k-smoothness for operators in both finite and infinite-dimensional Banach and Hilbert spaces, including specific space cases.

Findings

Characterization of k-smoothness in Hilbert and Banach spaces.

Identification of extreme contractions in certain operator spaces.

Extension of smoothness concepts to specific finite and infinite-dimensional spaces.

Abstract

We characterize $k-$smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize $k-$smoothness of operators on some particular spaces, namely $\mathbb{L}(\mathbb{X},\ell_{\infty}^n),~\mathbb{L}(\ell_{\infty}^3,\mathbb{Y}),$ where $\mathbb{X}$ is a finite-dimensional Banach space and $\mathbb{Y}$ is a two-dimensional Banach space. As an application, we characterize extreme contractions on $\mathbb{L}(\ell_{\infty}^3,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional polygonal Banach space.