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.
- the operator I+T has finite-dimensional kernel, and its image is closed and has finite codimension in X;
- 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;
- 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 lp
(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.
No prerequisites information is available yet.