Math 7334
Operator Theory
Instructor: Michael Loss
Lectures: TTh 1:35-2:55pm
Location:
Skiles 246
Office hours:
Tuesday 12-1:30 or by appointment
Textbook: Theory of linear
operators in Hilbert space by N.I. Akhiezer and I.M. Glazman,
Dover.
A
number
of
mathematical
and
physical
problems
can
be
formulated
and actually solved through the methods of operator theory.
The
prime example are linear evolution equations. They can be viewed as
initial value problems
involving linear operators on a Banach space or Hilbert space. To
stay with a concrete example, consider the heat equation on some
domain
with
some
boundary
conditions.
Associated
with
this
problem is a linear, unbounded operator, the generator of the heat
flow.
If
this operator is self-adjoint then it follows from the spectral
theorem that this evolution problem has a solution global in time.
The
notion
of
self-adjointness,
therefore,
captures
an
essential
part
of
the
nature of this problem, in particular features like boundary
conditions
must
enter
into
the
picture.
One
difficulty
is
that
in most of the interesting examples one has to deal with unbounded
operators.
Thus, the course will evolve as follows. We will review Hilbert
space theory, in particular the projection lemma, the Riesz
representation
theorem and the uniform boundedness principle. We will emphasize the
examples, such as standard L^2 theory, and maybe the Hilbert space
of
almost periodic functions. The next step up is to deal in detail
with bounded operators, in particular the theory of bounded
self-adjoint operators
culminating in the spectral theorem. There is a nice presentation in
Akhiezer and Glazman, however we shall take a different approach
using Banach
Banach and C* algebras and follow the treatment of Robert Zimmer.
The
fun starts with unbounded operators. The difference is now that the
domain is a defining part of the operator. The notion of a closed
operator is a relaxation
of the notion of continuity and is crucial for doing any kind of
analysis. The notion of self-adjointness is also tricky in this
context; there are symmetric operators that
are not self-adjoint and for which the spectral theorem does not
hold. These developments are not discussed for generalization's sake
but, as mentioned before, many of
the interesting
applications require unbounded operators. We will discuss via
examples some boundary value problems and their self-adjoint
extensions. The theory was
invented in large parts because of quantum
mechanics and if time permits we will talk about this topic.
I
will follow the textbook, which is a classic, more or less. This
textbook is good and cheap. Sometimes I will post additional notes
on this page. The notes will be
based on various books in addition to Akhiezer and Glazman:
Introductory
Functional Analysis with Applications, by Erwin Kreyszig,
Wiley.
Essential Results of
Functional Analysis, by Robert J. Zimmer, The University of
Chicago Press
Functional Analysis, by
Reed and Simon, Academic Press
Fourier Analysis and
Self-adjointness, Reed and Simon, Academic Press
Perturbation theory of
linear operators, by Tosio Kato, Springer
Linear Operators in
Hilbert Space, by Joachim Weidmann, Springer
Here
is a rough outline of the course with an approximate number of
lectures devoted to each topic:
Review of
Hilbert space theory: Projection lemma, Riesz representation
theorem, orthonormal systems and basis, examples of Hilbert spaces
such
as
the
space
of
almost
periodic
functions,
weak
and
strong convergence, weak compactness. about 3 Lectures
Linear
operators: Bounded operators, completely continuous
operators, projection operators, unitary operators with examples
such as the Fourier
transform. 3 Lectures
General concepts
for linear operators: Closed operators, invariant subspaces,
resolvent and spectrum, symmetric and selfadjoint operators,
with examples
such
as
multiplication
operators
and
differential
operators,
singular
integrals.
3 Lectures
Spectral analysis of bounded
operators: eigenspaces, Banach algebras and C* algebras,
spectral theorem for selfadjoint for bounded operators,
proof
of
the
fundamental
theorem
of
almost
periodic
functions, if time permits. 5 Lectures
Unbounded operators: Closed
operators, adjoint operators, range and kernel of unbounded
operators, fundamental
criterion for self-adjointness, Cayley transform,
spectral theorem for undbounded self-adjoint operators. 4
Lectures
Extension of
operators: symmetric operators versus selfadjoint
operators, deficiency indices, self-adjoint extensions,
semi-bounded operators,
Kato-Rellich theorem, examples . 5 Lectures
Semigroup
theory: If time permits, I will add a discussion about
semigroup theory, in particular the theorem of Hille-Yosida
concerning generators of
contraction semigroups. 4 Lectures.
GRADES:
There will be no tests but occasionally some homework which you are
required to hand in. The grade will be awarded according to
correctly solved
homework problems.
Summary on
Hilbert Space theory
L2 Spaces
Bounded
operators
A little note on
PDEs
General
Spectral Theory
Spectral
Theorem for bounded Self-adjoint Operators
Closed
operators
Basic
Theorem on Self-adjointness
The
Laplacian as a self adjoint operator
The
Kato-Rellich Theorem
The spectral theorem for unbounded
self adjoint operators
Extensions of symmetric
operators
The Theorem of
Hille and Yosida
Homework 1 (due
January 30, 2014, New deadline February 4)
Solutions for
Homework 1
Homework 2 (due
February 25)
Solutions for
Homework 2
Homework 3 (due
March 27)
Solutions for
Homework 3
Homework 4 (due April 15)
Solutions for
Homework 4