7280 Operator Theory - Spring 2006

University of Utah - Department of Mathematics
MATH 7280 Operator Theory- Spring 2006

Homework Assignments:
Homework # 1 (due February 17, 2006, at the end of the lecture)

Homework # 2 (due March 27, 2006, at the end of the lecture)

Homework # 3 (due April 24, 2006, at the end of the lecture)

Office: JWB 203
Phone: (801)585-6967
E-mail: bocea@math.utah.edu

Class meets: MWF 2:00 PM-2:50 PM. Room: JTB 110

Text: We will use selected chapters from the following books:

1. H. BREZIS, Analyse fonctionnelle. Theorie et Applications. Dunod, Paris, 1999
2. P. LAX, Functional Analysis. Wiley, 2002
3. F. HIRSCH and G. LACOMBE, Elements of Functional Analysis. Springer, 1999
4. R.E. EDWARDS, Functional Analysis: Theory and Applications. Dover, 1999

Course Web Page: http://www.math.utah.edu/~bocea/spring2006.html

Prerequisites: MATH 6210 Real Analysis, MATH 6220 Complex Analysis, or consent of instructor.

Course Audience: Graduate students interested in Functional Analysis and its applications.

Topics:
1. The Hahn-Banach theorem. Extensions of linear forms and separation of convex sets.
2. The Banach-Steinhaus, closed graph, and open mapping theorems. Unbounded operators, adjoint, the characterization of surjective operators.
3. Weak topologies. Reflexive, separable, and uniform convex spaces.
4. L^{p} spaces. Reflexivity, separability, duals, convolution and regularization.
5. Hilbert spaces. The theorems of Stampacchia and Lax-Milgram.
6. Compact operators. Spectral decomposition of selfadjoint compact operators. Riesz-Fredholm theory.
7. The Hille-Yoshida theorem. Maximal monotone operators.
8. The theory of distributions and Sobolev spaces. The variational formulation of boundary value problems for PDEs.

Homework and Final Exam: Homework will be assigned every 2-3 weeks. Homework assignments and their due dates will be posted on the web page of the course: http://www.math.utah.edu/~bocea/spring2006.html and announced in class as we proceed.
There will be a take home Final Exam. Details will be available on the web site of the course after April 15, 2006.

Grading:
Your final grade for the course will be determined as follows: Homework Average: 50%. Final Exam: 50%.