(Generalized) B-Fredholm theory relative to a Banach algebra homomorphism

We study two extensions of Harte’s Fredholm theory relative to a fixed Banach algebra homomorphism [2], namely B-Fredholm theory and generalized B-Fredholm theory (in short, GB-Fredholm theory), which are introduced and investigated in [1].

Following [2], with respect to a Banach algebra homomorphism T: A→B, an element a∈A is called Fredholm if Ta is invertible in B, and Weyl (resp. Browder) if it is the (resp. commuting) sum of an invertible element in A and an element in the null space of T. By replacing “invertible” by “Drazin invertible” (resp. “Koliha-Drazin invertible”), we obtain the notions of B-Fredholm, B-Weyl and B-Browder elements (resp. GB-Fredholm, GB-Weyl and GB-Browder elements) relative to a fixed Banach algebra homomorphism, which are studied in B-Fredholm theory (resp. GB-Fredholm theory) in general Banach algebras. We recall that an element a∈A is said to be Drazin invertible (resp. Koliha-Drazin invertible) if there exists an element b∈A such that ab=ba, b=bab and a−aba is nilpotent [3] (resp. quasinilpotent [4]). Clearly, invertible elements are Drazin invertible, which in turn are Koliha-Drazin invertible.

In this presentation, we will discuss several properties of (G)B-Fredholm, (G)B-Weyl and (G)B-Browder elements as well as the spectra derived from them.

References:

1. M. Cvetković, E. Boasso, and S. Živković-Zlatanović. Generalized B-Fredholm Banach algebra elements. Mediterr. J. Math., 13:3729–3746, 2016.
2. R. Harte. Fredholm theory relative to a Banach algebra homomorphism. Math. Z., 179:431–436, 1982.
3. M. Drazin. Pseudo-inverses in associative rings and semigroups. Amer. Math. Monthly, 65:506–514, 1958.
4. J. Koliha. A generalized Drazin inverse. Glasgow Math. J., 38:367–381, 1996.