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We study the Davis-Wielandt shell and the Davis-Wielandt radius of anoperator on a normed linear space $\mathcal{X}$. We show that after a suitablemodification, 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 normedlinear space and compute its value explicitly in case of some particularpolyhedral Banach spaces. We also present a general method to estimate theDavis-Wielandt index of any polyhedral Banach space.

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