Mathematics 511

Operator Theory and Applications

Department of Mathematics,  University of British Columbia

Instructor

Joel Feldman

Prerequisites

• A course in measure theory at the level of UBC's Math 420/Math 507.
• It would be desirable to have also taken a course on Hilbert and/or Banach spaces like UBC's Math 421/Math 510, but this is not essential. I will adjust the level of the course according to what proportion of the class have taken such a course.

Text

• Michael Reed and Barry Simon, Functional Analysis (Methods of modern mathematical physics, volume 1, Academic Press, 1980). This is an excellent book, but it is also unconscionably expensive. So I have not made it a required textbook.

I will post all handouts, problem sets, etc. on the web here.

Topics

1. Review of Hilbert and Banach Spaces:
    Definitions, examples, elementary geometry
    Operators - linear, bounded, compact, hermitian, self-adjoint, unitary
2. The Spectral Theorems:
    I will state several versions of the spectral theorem. The extent to which this is treated as a review, and in particular how many proofs I give, will depend on what proportion of the class has already seen rigorous proofs of one or more variants of the spectral theorem.
3. Unbounded Operators:
    Examples
    Closed and preclosed unbounded operators
    Self--adjoint extensions of hermitian operators
    Bloch theory and the spectrum of periodic Schrödinger operators

Grading

• The grade will be based on regular problem sets.

Policies

• Missing a homework normally results in a mark of 0. Exceptions may be granted in two cases: prior consent of the instructor or a medical emergency.