We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

中文翻译：

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加权核心-EP 逆和加权核心-EP 在-代数中的预购

我们定义了在希尔伯特空间上的有界线性算子的加权核心-EP 逆和加权核心-EP 预阶到 a 的元素的扩展。 $C^{\ast }$ -代数。加权核-EP 逆和加权核-EP 预序的一些性质被推广并证明了一些新性质。使用加权元素、加权核心-EP预序、负偏序和某些元素的星偏序，在所有元素的集合上呈现新的加权预序 $wg$ -Drazin 可逆元素 $C^{\ast }$ -代数。应用这些结果，我们引入并表征了新的偏序，它们将核心-EP 预序扩展为偏序。