Course Announcement for Fall 2004
Math 528A Banach and Hilbert Space
MWF 10:00-10:50 AM in Math 501
Contact: William Faris, Math 620, faris@math.arizona.edu
This course will concentrate on the spectral theory of linear
operators acting in Hilbert space. The text consists of notes that
will be available at the beginning of the semester.
A landmark result in matrix theory is the spectral theorem for
self-adjoint matrices. The generalization to the Hilbert space setting
is the spectral theorem for self-adjoint operators.
Matrix Theorem: Every self-adjoint matrix is unitarily equivalent to a
real diagonal matrix.
Operator Theorem: Every self-adjoint operator is unitarily equivalent
to a real multiplication operator on a space of square-integrable
functions.
The unitary equivalence is given by a unitary operator. In the matrix
case this is a unitary matrix given by an orthonormal basis of
eigenvectors. In the operator case the situation is more subtle, but
there is an analogous representation.
The goal is to understand the spectral theorem and its
consequences. Examples will illustrate the different kinds of spectra.
Spectral theory is infinite dimensional linear algebra, and so it is
useful in many branches of mathematics, science, and
engineering. However it plays a special role in quantum physics, where
the basic equations are dictated by a commutation relation between
self-adjoint operators. The application to quantum physics will be the
focus of Math 527B in the second semester.