Regular Fredholm Pairs

In this work it is introduced the notion of regular Fredholm pair, i.e. a Fredholm pair whose operators are regular. The main properties of these objects are studied, and what is more, they are entirely classified. Furthermore, the index of a Fredholm pair turns out to be an extremely useful tool in the description of the aforementioned objects. Finally, regular Fredholm pairs are characterized in terms of regular Fredholm symmetrical pairs, exact chains of multiplication operators, and invertible Banach space operators.