MAGIC043: Banach spaces and their operators

The course studies Banach spaces and operators acting on them, thus providing an introduction to an important branch of modern infinite-dimensional linear analysis. To be precise, the starting point of the course is the following classical theorem of F. Riesz. Let T be a compact operator on a Banach space X, and let I be the identity operator on X. Then:

1. the operator I+T has finite-dimensional kernel, and its image is closed and has finite codimension in X;
2. there is a non-negative integer n such that the kernel of (I+T)ⁿ is equal to the kernel of (I+T)ⁿ⁺¹ and the image of (I+T)ⁿ is equal to the image of (I+T)ⁿ⁺¹;
3. each non-zero point of the spectrum of T is an eigenvalue for T, and 0 is the only possible accumulation point of the spectrum of T.

The first part of the course is devoted to the study of these properties and their interrelationship, starting from a purely algebraic viewpoint. In the second part of the course, we shall introduce the concept of a Schauder basis for a Banach space. This is the natural analogue of an orthonormal basis for a Hilbert space, or a Hamel basis for a vector space; note, however, that in contrast to these examples, a Banach space may not have a Schauder basis. As an application of Schauder bases we shall prove that the ideal of compact operators is the only non-trivial closed ideal in the ring of all bounded linear operators on each of the classical sequence spaces l_p (for 1 ≤ p < ∞) and c₀; this result is due to Calkin (1941) for p=2 and to Gohberg, Markus, and Feldman (1960) in the general case.
Outline of syllabus: Index theory for Fredholm operators; Riesz operators and inessential operators; Schauder bases in Banach spaces; Gohberg-Markus-Feldman's characterization of the closed ideals in the classical sequence spaces.

1. Course outline and motivation; background results from infinite-dimensional linear algebra.
2. The Index Theorem for Fredholm mappings.
3. Linear mappings with finite ascent and finite descent.
4. Brief review of fundamental background results from functional analysis; operator ideals.
5. Introduction to Fredholm operators and semi-Fredholm operators.
6. Yood's Lemma and Atkinson's Theorem.
7. Continuity of the Fredholm index.
8. Riesz-Schauder operators; introduction to Riesz operators.
9. The holomorphic function calculus and Riesz' Idempotent Theorem.
10. Riesz operators and the essential spectrum.
11. Inessential operators.
12. The Jacobson radical and Kleinecke's characterization of the inessential operators.
13. Strictly singular operators.
14. Introduction to Schauder bases in Banach spaces.
15. Characterizations and properties of Schauder bases.
16. Unconditional Schauder bases.
17. Equivalence and stability of Schauder bases.
18. Block basic sequences and Bessaga-Pelczynski's Selection Principle.
19. Setting the stage for Gohberg-Markus-Feldman's Theorem: the standard bases of the classical sequence spaces l_p (1 ≤ p < ∞) and c₀.
20. The proof of Gohberg-Markus-Feldman's Theorem.