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

TL;DR

This paper extends the Davis-Wielandt shell concept to semi-Hilbertian space operators, exploring their properties, parallelism, and radii, thereby generalizing existing results in operator theory.

Contribution

It introduces a generalized Davis-Wielandt shell for semi-Hilbertian operators and characterizes $A$-normaloid operators using $A$-Davis-Wielandt radii, extending prior work.

Findings

Characterization of $A$-normaloid operators via $A$-Davis-Wielandt radii

Connection between $A$-seminorm-parallelism and $A$-Davis-Wielandt radius

Generalization of known results in operator theory

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 \cite{zamanilma2018,chanchan}. Some other related results are also discussed.