Algebraic Fredholm Theory

Joel H. Shapiro


Lecture Notes 2011

Abstract: These notes develop the purely algebraic parts of the basic theory of Fredholm operators. The setting is vector spaces over arbitrary fields, and linear transformations between these spaces. No topology is assumed for the vector spaces and consequently no continuity is assumed for the linear transformations.  Within this reduced setting the notion of ``Fredholm transformation'' is studied, emphasizing invertibility properties, expression of these properties in terms of Fredholm index, invariance of these properties under finite-rank perturbations, and implications for the study of spectra. The exposition culminates in a Fredholm-inspired analysis of the spectra of multiplication operators on spaces of holomorphic functions.

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