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General


Description

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 course is devoted to the study of these properties and their interrelationship, starting from a purely algebraic viewpoint.

Semester

Spring 2018 (Monday, January 22 to Friday, March 16; Monday, April 23 to Friday, May 4)

Timetable

  • Thu 14:05 - 14:55

Prerequisites

Basic knowledge of functional analysis, up to and including the Open Mapping Theorem and the Hahn-Banach Theorem; MAGIC061 covers this, and much more.

Syllabus

  1. Course outline and motivation; background results from infinite-dimensional linear algebra; the Index Theorem for Fredholm mappings.
  2. Linear mappings with finite ascent and finite descent.
  3. Brief review of fundamental background results from functional analysis; operator ideals.
  4. Introduction to Fredholm operators and semi-Fredholm operators.
  5. Yood's Lemma and Atkinson's Theorem.
  6. Continuity of the Fredholm index.
  7. Riesz-Schauder operators; introduction to Riesz operators.
  8. The holomorphic function calculus and Riesz' Idempotent Theorem.
  9. Riesz operators and the essential spectrum.
  10. Inessential operators, the Jacobson radical and Kleinecke's characterization of the inessential operators.

Lecturer


Niels Laustsen
Email n.laustsen@lancaster.ac.uk
Phone (01524) 594617
Interests Operators on Banach spaces, Banach algebras, K-theory
Photo of Niels Laustsen


Students


Photo of Maxime Fairon
Maxime Fairon
(Leeds)


Bibliography


An introduction to Banach space theoryR. E. Megginson
Introduction to functional analysisR. Meise and D. Vogt


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