Generalized B-Fredholm Banach algebra elements

Given a (not necessarily continuous) homomorphism between Banach algebras $\text{T}\text{\hspace{0.17em}⁣}:\text{A}\to \text{B}$, an element $a\in \text{A}$ will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to $\text{T}$, if $\text{T}(a)\in \text{B}$ 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.