ABSTRACT
Chapter 8 is focused on the Fredholm theory and Fredholm operators which are generalizations of operators that are the difference of the identity and a compact linear operator on a Banach space. Fredholm operators play a very important role in the spectral theory of operators. The chapter presents a study of Fredholm and semi–Fredholm operators, the index and Atkinson’s theorems, and Yood’s results for all and Yood’s results for all upper semi–Fredholm operators with nonpositive index, lower semi–Fredholm operators with nonnegative index, and properties of right and left Fredholm operators. It also establishes the openness of the set of proper semi–Fredholm operators in the space of bounded linear operators between Banach spaces, and gives the proofs of the punctured neighborhood theorem and the Kato decomposition theorem, for which West’s proof is used. Finally it provides detailed studies of the ascent and descent of operators, and of the generalized kernel and range of bounded linear operators, the properties of Browder and semi–Browder operators, essential spectra and essential type subsets of the spectrum.
