R-Fredholm theory in ordered Banach algebras

Abstract

The introduction of Fredholm theory relative to general unital homomorphisms $$T:A \rightarrow B$$ between Banach algebras A and B, which involves the study of Fredholm, Weyl and Browder elements, was due to Harte (Math Z 179:431–436, 1982), after which this investigation was continued by several authors. Motivated by results of Alekhno (Positivity 11:375–386, 2007, Positivity 13:3–20, 2009), the definitions of upper Weyl and upper Browder elements in an ordered Banach algebra (OBA) were given in Benjamin and Mouton (Quaest Math 39:643–664, 2016), thereby initiating the study of the interplay between Fredholm theory and ordering. Inspired by an indication that these elements could be useful in the context of OBAs, certain variants of the invertible, Fredholm, Weyl and Browder elements (namely the r-invertible, r-Fredholm, r-Weyl and r-Browder elements) were investigated in Benjamin et al. (Glasgow Math J, 2018. https://doi.org/10.1017/S0017089518000393) in the context of general Banach algebras. Continuing the development of Fredholm theory in ordered Banach algebras, we now introduce the notions of upper r-Weyl, contractive upper r-Weyl, upper r-Browder and contractive upper r-Browder elements in an OBA and, motivated by previous results, investigate conditions under which the positive almost r-invertible r-Fredholm elements will be contractive upper r-Browder. This also leads to the recovering (and strengthening) of certain results in Benjamin and Mouton (Positivity 21:575–592, 2017) regarding the upper Browder spectrum property.