LAUSR

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 Ivkovic (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), Djordjevic (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 Ivkovic (Banach J Math Anal 13(4):989–1016, 2019; Ann Funct Anal, 2020. https://doi.org/10.1007/43034-019-00034-z ).