Starting from the definition of \({\mathcal {A}}\)-Fredholm and semi-\({\mathcal {A}}\)-Fredholm operator on the standard module over a unital \(C^{*}\) algebra \({\mathcal {A}}\), introduced in Ivković (Banach J Math Anal 13(4):989–1016, 2019) and Mishchenko and Fomenko (Izv Akad Nauk SSSR Ser Mat 43:831–859, 1979), we construct various generalizations of these operators and obtain several results as an analogue or a generalization of the results in Berkani and Sarih (Glasg Math J 43(3):457–465, 2001. https://doi.org/10.1017/S0017089501030075), Berkani (Proc Am Math Soc 130(6):1717–1723, 2001), Djordjević (Proc Am Math Soc 130(1):81–84, 2001) and Yang (Trans Am Math Soc 216:313–326, 1976). Moreover, we also study non-adjointable semi-\({\mathcal {A}}\)-Fredholm operators as a natural continuation of the work in Irmatov and Mishchenko (J K-Theory 2:329–351, 2008. https://doi.org/10.1017/is008004001jkt034) on non-adjointable \({\mathcal {A}}\)-Fredholm operators and obtain an analogue or a generalization in this setting of the results in Ivković (Banach J Math Anal 13(4):989–1016, 2019; Ann Funct Anal, 2020. https://doi.org/10.1007/43034-019-00034-z).

中文翻译：

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关于半$$ {\ mathcal {A}} $$ A -Fredholm运算符的各种推广

从\（{\ mathcal {A}} \）- Fredholm和Semi- \（{\ mathcal {A}} \）- Fredholm运算符的定义开始，在标准模块上的\（C ^ {*} \ ）代数\（{\ mathcal {A}} \），分别在Ivković（Banach J Math Anal 13（4）：989–1016，2019）和Mishchenko和Fomenko（Izv Akad Nauk SSSR Ser Mat 43：831–859，1979）中引入，我们对这些算子进行了各种归纳并获得了几种结果作为Berkani和Sarih（Glasg Math J 43（3）：457–465，2001.https：//doi.org/10.1017/S0017089501030075），Berkani（Proc Am Math Soc 130（ 6）：1717–1723，2001），Djordjević（Proc Am Math Soc 130（1）：81–84，2001）和Yang（Trans Am Math Soc 216：313–326，1976）。此外，我们还研究了非可连接的半\（{{mathcal {A}} \）- Fredholm运算符，作为Irmatov和Mishchenko的工作的自然延续（J K-理论2：329-351，2008。https： //doi.org/10.1017/is008004001jkt034）在不可连接的\（{\ mathcal {A}} \）上-Fredholm运算符，并在Ivković的这种结果设置中获得类似物或概括（Banach J Math Anal 13（4）：989-1016，2019; Ann Funct Anal，2020. https://doi.org/10.1007/ 43034-019-00034-z）。