Generalized B-Fredholm Banach algebra elements

Given a (not necessarily continuous) homomorphism between Banach algebras T:A→B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{T} : \mathcal{A} \to \mathcal{B}}}$$\end{document}, an element a∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \in {\mathcal{A}}}$$\end{document} will be said to be B-Fredholm (respectively, generalized B-Fredholm) relative to T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{T}}$$\end{document}, if T(a)∈B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{T}(a) \in \mathcal{B}}}$$\end{document} is Drazin invertible (respectively, Koliha–Drazin invertible). In this article, the aforementioned elements will be characterized and their main properties will be studied. In addition, perturbation properties will be also considered.