MATH 712: Compact Operators on Hilbert Space (Spring 2004)

Instructor: Todd Kemp

Meeting Time & Room

We will give a detailed treatment of the theory of compact operators on Hilbert space. After some preliminary basic functional analysis (mostly review from MATH 611), we will prove the spectral theorem for compact self-adjoint operators, and discuss its extension to normal operators. We will then discuss Fredholm theory, followed by some of the theory of Volterra integral operators (which are non-self-adjoint and fail the spectral theorem).

After this, time permitting, we will discuss the following more modern topics in some detail:

• Introduction to nonselfadjoint theory; the singular value theorem
• Operator norms and operator spaces
• Hilbert-Schmidt operators and nuclear (trace-class) operators
• Schatten-von Neumann ideals and non-commutative L^p spaces
• Classification of ideals

The only prerequisite for this course is MATH 611. MATH 612 would be a useful corequisite, but is not absolutely required.

The recommended textbook for the first part of the course is Friedman, Avner: Foundations of Modern Analysis. Dover Publications, New York, 1982.

For the latter topics, course notes will be provided by the instructor.

Last modified: November 18, 2003