In this paper, we consider \({\mathcal A}\)-Fredholm and semi-\({\mathcal A}\)-Fredholm operators on Hilbert C*-modules over a W*-algebra \({\mathcal A}\) defined in [3] and [9]. Using the assumption that \({\mathcal A}\) is a W*-algebra (rather than an arbitrary C*-algebra), we obtain a generalization of Schechter—Lebow characterization of semi-Fredholm operators and a generalization of the “punctured neighborhood” theorem, as well as some other results generalizing their classical counterparts. We consider both adjointable and nonadjointable semi-Fredholm operators over W*-algebras. Moreover, we also work with general bounded adjointable operators with closed ranges over C*-algebras and prove a generalization of a Bouldin result for Hilbert spaces to Hilbert C*-modules.

中文翻译：

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W *-代数上的带近距离算子和半弗雷德霍姆算子

在本文中，我们考虑W *-代数\ { {\ mathcal A}上Hilbert C *-模上的\（{\ mathcal A} \）- Fredholm和半\\ {{\ mathcal A} \}- Fredholm算子} \）在[3]和[9]中定义。使用\（{\ mathcal A} \）是W *-代数（而不是任意C *-代数）的假设，我们得到了半弗雷德霍姆算子的Schechter-Lebow推广和“邻域定理”，以及推广其经典对应物的其他一些结果。我们考虑W上的可连接和不可连接的半弗雷德霍姆算子*-代数 此外，我们还使用C *-代数上具有封闭范围的一般有界可邻接算子，并证明了将希尔伯特空间的Bouldin结果推广到希尔伯特C *-模。