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 X . We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on L(X) which is equivalent to the usual operator norm on L(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 finite-dimensional Banach space.

Refinements of the Euclidean Operator Radius and Davis–Wielandt Radius-Type Inequalities

“…Remark 1. All obtained results of Section 3 are valid for the generalized Euclidean operator radius ω p (•) (p ≥ 1) by using (12) instead of (13). We leave the rest of the generalizations to the interested reader.…”

“…For further inequalities of the Euclidean operator radius combined with several basic properties, the reader may refer to [3,4,6,9,10]. For more generalization, counterparts, and recent related results, the reader may refer to [11][12][13][14][15][16][17][18][19].…”

A Generalized Norm on Reproducing Kernel Hilbert Spaces and Its Applications

“…Additionally, we can observe that T 0,1 = ω(T) and T 1,0 = T , taking α = 0, β = 1, and α = 1, β = 0, respectively, yield the same formulas as for the numerical radius and operator norm. Another notable instance of the (α, β)-norm occurs when we choose α = β = 1, which leads to the modified Davis-Wielandt radius of the operator T ∈ L(E ), denoted by dω * (T) (see [3]). In [2], it was established that:…”

Numerical radius inequalities of operator matrices from a new norm on B(H)

“…We note that if α = 1, β = 0 then T α,β = w(T ), and if α = 0, β = 1 then T α,β = T . Also, if we consider α = β = 1, then we have the modified Davis-Wielandt radius of T , that is, T α,β = dw * (T ), (see [9]). In this article, we consider α + β = 1, i.e., β = 1 − α and explore the α-norm of n × n operator matrices, where the α-norm of T is defined as:…”