On the Davis-Wielandt shell of an operator and the Davis-Wielandt index of a normed linear space

TL;DR

This paper investigates the Davis-Wielandt shell and radius of operators in normed spaces, introduces a related index, and provides explicit calculations and estimation methods for polyhedral Banach spaces.

Contribution

It introduces the Davis-Wielandt index for normed spaces and establishes a new norm on operators that is equivalent to the standard operator norm.

Findings

Modified Davis-Wielandt radius defines an equivalent norm on operator space.

Explicit Davis-Wielandt index values for certain polyhedral Banach spaces.

A general method to estimate the Davis-Wielandt index for polyhedral Banach spaces.

Abstract

We study the Davis-Wielandt shell and the Davis-Wielandt radius of an operator on a normed linear space $\mathcal{X}$. We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on $\mathcal{L}(\mathcal{X})$ which is equivalent to the usual operator norm on $\mathcal{L}(\mathcal{X})$. We introduce the Davis-Wielandt index of a normed linear space and compute its value explicitly in case of some particular polyhedral Banach spaces. We also present a general method to estimate the Davis-Wielandt index of any polyhedral Banach space.