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

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 Chan and Chan (Oper Matrices 11(3):885–890, 2017), Zamani et al. (Linear Multilinear Algebra 67(11):2147–2158, 2019). Some other related results are also discussed.