Syllabus
1. Fredholm operators: Definitions and basic properties, duality, Yood's Lemma and its consequences, Atkinson's Theorem, continuity of the Fredholm index, Riesz-Schauder operators.
2. Riesz operators: Definition, algebraic properties, Interlude (Banach algebras, spectral theory, holomorphic function calculus), the essential spectrum.
3. Inessential operators: Definition and basic properties, the Jacobson radical, Kleinecke's Theorem.
4. Strictly singular operators: Introduction, Kato's Lemma, the strictly singular operators form an operator ideal, the strictly singular operators contain the compact operators.
5. Schauder bases: Introduction, the basis projections and the coordinate functionals, basic sequences, duality, equivalence of bases and stability, block basic sequences and the Bessaga-Pelczynski Selection Principle.
6. The Gohberg-Markus-Feldman Theorem: Applying the B-P Selection Principle to c_0 and l_p, the proof of the G-M-F Theorem.
7. Separable C(K) spaces: Revision (dual space, extreme points, separability, Banach-Stone, Stone-Weierstrass, containment of c_0) the Cantor set, universality of C([0,1]), Miljutin theorem. Countable compacta: the dual is l_1, C(K) is c_0 saturated, hints at the classification, there are uncountably many non isomorphic C(K) spaces
8. Tentative: Eidelheit's Theorem, the structure of homomorphisms from B(X) and A(X).
Annotation
The main aim of this course is to study
Fredholm, Riesz, strictly singular, and inessential operators on Banach spaces. As a tool we additionally introduce and study Schauder bases in Banach spaces.