Math 7334
Operator Theory
Instructor: Michael Loss
Lectures: TTh 9:35-10:55
Location:
Skiles 154
Office hours:
Tuesday 12 -1, Thursday 1-2 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 is that certain evolution equations can be viewed as
an initial value problem of an `ordinary differential equation'
involving a linear operator 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 selfadjoint then it follows from the spectral
theorem that this evolution problem has a solution global in time.
The
notion
of
selfadjointness,
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, as well as the Hilbert space
of
almost periodic functions. We will then talk about linear operators,
at first only bounded ones and work out the spectral theory of
completely continuous
operators
in
detail.
These
are
important
since
they
are
a
direct generalization of matrices. Many practical problems can be
reduced to this case.
As
an
application
we
use
this
theory
to
prove
the
fundamental theorem of H. Bohr on almost periodic functions.
The next step up is to deal in detail with bounded
operators, in particular the theory of bounded selfadjoint operators
culminating in the spectral
theorem. Here the notion of operator valued function becomes
important.
The
fun starts with unbounded operators. The difference is now that the
domain is a defining part of the operator. The notion of closed is a
relaxation
of
the
notion
of
continuity
and
is
crucial
for
doing
any kind of analysis. The notion of selfadjointness is also tricky;
there are symmetric operators that
are
not
selfadjoint
and
for
which
the
spectral
theorem
does
not hold. These developments are not discussed for generalization's
sake but many of the interesting applications
require unbounded operators. 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,
very closely. This textbook is good and cheap. Sometimes I will post
additional notes here.
Here is a rough syllabus 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. 4 Lectures
Linear
operators: Bounded operators, completely continuous
operators, projection operators, unitary operators with examples
such as the Fourier
transform. 4 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.
4
Lectures
Spectral analysis of completely
continuous operators: eigenspaces and eigenvalues, spectral
theorem for selfadjoint completely continuous operators,
proof
of
the
fundamental
theorem
of
almost
periodic
functions. 4 Lectures
Spectral
analysis of unitary and selfadjoint operators: Bochner's
theorem, spectral theorem for unitary operators, spectral theorem
for unitary operatorrs, Cayley
transforms, examples. 5 Lectures
Extension of
operators: Adjoint operators, symetric operators versus
selfadjoint operators, deficiency indices, semi-bounded operators. 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.
Homework 1, Due February 7.
Notes: Neumann
Series
Homework 2, Due
March 6
Fredholm Alternative
Homework 3, Due April 4
Eigenvalues for Compact
Operators
Homework
4, due April 24
Basic theorem on
self-adjointness
Notes on isometries
Self-adjoint extensions
A
summary on the Fourier transform
The
Laplacian
The Kato-Rellich Theorem