Graduate Course 3341.621
Operator Theory
Spring 2010
Instructor: Woo Young Lee
Office: 27-425
Phone: 02-880-1435
Course Meeting Schedule: Mon/Wed 14:00-15:15 (25-110)
Office Hours: Mon/Wed 11:00-12:00 (27-425)
Teaching Assistant: Dong-O Kang
Course Goals:
The goal of this course is to set forth some of the recent developments that had taken place in Operator Theory.
In particular we focus on the Fredholm theory, Weyl theory, hyponormal and subnormal theory; weighted shifts theory,
Toeplitz theory; and the invariant subspace problem. This course is appropriate and useful for graduate students in mathematics.
Prerequisites: A good course in Functional Analysis
Course Books:
# Text: W. Y. Lee, Lectures on Operator Theory, Lecture Note, SNU, 2010.
(Clicking here leads to the Lecture Notes.)
# Recommended:
- J. Conway, The theory of subnormal operators, Math. Surveys Monographs 36, AMS, 1991
- R. Douglas, Banach algebra techniques in operator theory, Springer, 1998
- I. Gohberg, S. Goldberg, M. Kaashoek, Classes of linear operators, Vol 1, Birkhauser, 1990
- S. Goldberg, Unbounded linear operators, McFraw-Hill, 1966
- P. Halmos, A Hilbert space problem book, Springer, 1974
- R. Harte, Invertibility and singularity, Dekker, 1988
- G. Murphy, C*-algebras and operator theory, Academic Press. 1990
- C. Pearcy, Some recent developments in operator theory, CBMS 36, AMS, 1978
Examination and Grading:
The course grade will be computed by weighing one midterm research report 30%, the final research report 30%,
and the discussion section grade (based on presentations, assignments and in-class performance) will count 40%.
Research reports: Instead of written examination, there will be two research reports.
You will have to write a research report on one or several of the given topics - essentially, open problems, within each due date.
Only one topic is highly recommended for each project. Please try to completely solve it;
otherwise you might give a nice partial answer. The grade of the research report is based on
the completeness, the clearness, the importance, and the elegance of the main result.
|
Project |
Due Date |
Topics (from the Lecture Notes) |
|
Midterm |
May 3 |
Problems 1.1, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2 and Conjecture 3.3 |
|
Final |
June 16 |
Problems 4.1, 4.2, 4.3, 4.4, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11 5.1, 5.2, 5.3, 5.5, 5.6, 5.10, 5.11 |
Detailed Syllabus
|
Week |
Date |
Sections |
Dues |
|
1 |
3/4 |
Preliminaries I |
3/1 Holiday |
|
2 |
3/8, 3/10 |
Preliminaries II |
|
|
3 |
3/15, 3/17 |
Preliminaries III |
|
|
4 |
3/22, 3/24 |
Definitions and examples, Operators with closed ranges, The product of Fredholm operators, Perturbation theorems |
|
|
5 |
3/29, 3/31 |
The Calkin algebra, The punctured neighborhood theorem |
|
|
6 |
4/5, 4/7 |
The Riesz-Schauder theory, Essential spectra |
|
|
7 |
4/12, 4/14 |
The continuity of spectra, Weyl’s theorem |
|
|
8 |
4/19, 4/21 |
Spectral mapping theorem for the Weyl spectrum,Perturbation theorems, Hyponormal operators |
|
|
9 |
4/26, 4/28 |
The Berger-Shaw theorem, Subnormal operators |
|
|
10 |
5/3 |
Cyclic vectors, p-hyponormal operators |
5/3: Due for the midterm 5/5 Holiday |
|
11 |
5/10, 5/12 |
Berger’s theorem, k-hyponormality |
|
|
12 |
5/17, 5/19 |
Fourier transform and Beurling’s theorem, Hardy space |
|
|
13 |
5/24, 5/26 |
Toeplitz operators |
|
|
14 |
5/31, 6/1 |
Hyponormality of Toeplitz operators |
|
|
15 |
6/7, 6/9 |
Subnormality of Toeplitz operators |
6/9: Due for the final |