Here is a rough outline of my Spring 2001 course Introduction to Operator Theory MATH-7334 - Review of some basic principles of Hilbert and Banach space geometry.
Norms, completeness, continuous linear maps, scalar product. Duality in Banach spaces, dual operators, Hahn-Banach theorem. Closed graph theorem, Banach Steinhaus theorem (uniform boundedness principle). Spectrum of a bounded operator, examples. Literature: W. Rudin, F. Riesz, R. Zimmer. - Compact operators.
Compact sets in metric spaces, Arcela-Ascoli theorem, compact sets in L p Spectral theory of compact operators (Riesz-Schauder Theory). Selfadjoint bounded operators, Hilbert-Schmidt operators. Application to compact groups: Basics concepts of representation theory, Haar measure, Schur's lemma. The Peter-Weyl theorem. Classification of unitary representations for SU(2) analytic approach. Decompositions of L (G/K), G a compact group and K closed subgroup, spherical harmonics on S and S Fredholm's alternative, compactness of weakly singular operators. Application to the classical Dirichlet problem for the Laplacian (with "smooth" boundary), potential theory approach.
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