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Fredholm, Weyl and Browder theory. (English) Zbl 0567.47001
If $$K: X\to X$$ is a compact operator on a normed space X then $$T=I-K$$ is ”Fredholm” in a number of different ways: it has a ”generalized inverse” $$S_ 1$$ for which $$T=T S_ 1T$$; it has an ”essential inverse” $$S_ 2$$ for which $$I-S_ 2T$$ and I-T $$S_ 2$$ are both compact; it has a ”spectral projection” I-P and a ”Drazin inverse” for which $$P^ 2=P=S_ 3T=T S_ 3$$ and T(I-P) is nilpotent. In this account we seek to expose the elementary algebraic links between these three ideas, working in the framework of an additive category, and hence to set the classical theory of compact operators in an algebraic setting.

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A53 (Semi-) Fredholm operators; index theories