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

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.

References in corpus (4)

• Recent progress and open questions on the numerical index of Banach spaces
• On the numerical index of polyhedral Banach spaces
• Bounds for the Davis-Wielandt radius of bounded linear operators
• On numerical radius and Crawford number attainment sets of a bounded linear operator