Fredholm complements of upper triangular operator matrices

Qiu, Sinan; Jiang, Lining

doi:10.1007/s43037-024-00340-2

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2024

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“…Fredholm theory originates from the problem of solving integral equations and plays a significant role in operator theory. The classical Fredholm theory can be built in compact manifolds as the study of Fredholm sections of some Banach bundle over the Banach manifold [1]. As we have known, Atkinson established a connection between Fredholm operators and the invertible elements in Calkin algebra [2], which indicates that the Fredholm operators can be regarded as "weakened invertible operators".…”

Section: Introductionmentioning

confidence: 99%

Fredholm Theory Relative to Any Algebra Homomorphisms

Kong,

Wang,

Yang

2024

Mathematics

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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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Section: Introductionmentioning

confidence: 99%