On various generalizations of semi-\(\mathcal{A}\)-Fredholm operators (Q2181143)

Let \(\mathcal{A}\) a unital \(C^*\)-algebra and \(H_{\mathcal A}\) be the standard module \(\mathcal{A}\). \textit{A. S. Mishchenko} and \textit{A. T. Fomenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831--859 (1979; Zbl 0416.46052)] introduced and studied the notion of an \(\mathcal{A}\)-Fredholm operator on \(H_{\mathcal A}\). The author [Banach J. Math. Anal. 13, No. 4, 989--1016 (2019; Zbl 1440.47008)] defined and studied semi-\(\mathcal{A}\)-Fredholm operators on \(H_{\mathcal{A}}\). In the present paper, he goes further and defines generalized \(\mathcal{A}\)-Fredholm operators, generalized \(\mathcal{A}\)-Weyl operators, and semi-\(\mathcal{A}\)-\(\mathcal{B}\)-Fredholm operators on \(H_{\mathcal{A}}\). Several properties of these operators are obtained, which extend several known results.