Davis–Wielandt shells of semi-Hilbertian space operators and its applications

Abstract

In this paper we generalize the concept of Davis-Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator A is considered. Moreover, we investigate the parallelism of A-bounded operators with respect to the seminorm and the numerical radius induced by A. Mainly, we characterize A-normaloid operators in terms of their A-Davis-Wielandt radii. In addition, a connection between A-seminorm-parallelism to the identity operator and an equality condition for the A-Davis-Wielandt radius is proved. This generalizes the well-known results in [23,10]. Some other related results are also discussed.

A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications

“…T = {λ ∈ C : |λ| = 1}. Recently, new types of parallelism for A-bounded operators based on the A-numerical radius and the A-operator seminorm was introduced in[15]. More precisely, we have the following definition.…”

Some $\mathbb{A}$-numerical radius inequalities for $d\times d$ operator matrices

“…Recently, many results covering some classes of operators on a complex Hilbert space H, • | • are extended to H, • | • A (see, e.g., [13,12,17,5,6,20,16]).…”

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

“…For recent articles about the Davis-Wielandt shell and the Davis-Wielandt radius of an operator, we refer to see [2,5]. The above concepts are closely related to the numerical range of the operator T, studied initially by Bauer [1] and Lumer [8], in the more general setting of normed linear spaces.…”