Abstract
In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms.
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