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.

中文翻译：

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半希尔伯特空间算子的 Davis-Wielandt 壳及其应用

在本文中，当考虑由正算子 $A$ 引起的半内积时，我们在 Hilbert 空间上推广了算子的 Davis-Wielandt 壳的概念。此外，我们研究了 $A$ 有界算子相对于 $A$ 引起的半范数和数值半径的并行性。主要是，我们根据 $A$-Davis-Wielandt 半径来表征 $A$-normaloid 算子。此外，证明了$A$-seminorm-parallelism 与恒等运算符之间的联系以及$A$-Davis-Wielandt 半径的相等条件。这概括了\cite{zamanilma2018,chanchan} 中众所周知的结果。还讨论了一些其他相关结果。