  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.
