Math 7334 Operator Theory
Instructor: Michael Loss
Lectures: MWF 11:05-11:55 am
Location:
Skiles 268
Office hours: MW
14:05-14:55 or by appointment
Textbook:
Functional Analysis: An Introduction, by
Yuri Eidelman, Vtali Milman and Antonis Tsolomitis
Graduate Studies in Mathematics Vol 66, American Mathematical
Society. (EMT)
The topics of this course are unbounded operator theory, and an
introduction to operator algebras. Both are very much related
to the mathematical foundation of quantum mechanics, mainly through
the work of von Neuman, but are used in various other mathematical
and physical disciplines.
Certain linear evolution problems can be formulated and actually
solved through the methods of operator theory.
They
can be viewed as initial value problems involving linear operators
on a Banach space or Hilbert space. As 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,
where the operator is determined in part by its domain of
definition. While the notion of a symmetric operator is quite
natural it is not sufficient
for the spectral theorem to hold; more is needed. We have to
understand the notion of self-adjointness and somehow classify
extension
of
symmetric operators.
The
goal for this first part of the course is to develop an
understanding of these concepts and to apply them to a some
non-trivial problems in partial differential
equations.
Our next topic will be operator algebras. It is a fruitful way
instead of considering bounded linear operators on a Hilbert space,
to look at them in an abstract fashion, i.e.,
in terms of normed rings or what is the same normed algebras. Again,
this approach was pioneered by von Neuman and later developed by
Gelfand, Naimark and
many
others. Many topics become quite transparent, e.g., the spectral
theorem can be understood as classification of commutative C*
algebras (Gelfand_Naimark theorem).
Again Gelfand and Naimark pioneered the theory of C* algebras, which
are (in general non-cummative) Banach algebras
with
the uniform topology and an abstract notion of an adjoint and the .
In quantum mechanics observables form C* algebras and one aims at
studying them and their
representations. If one considers such algebras with the weak
topology one deals with von Neuman Algebras.
The learning goal for this section is to gain some familiarity
with some of these concepts and have some understanding of the
basic theorems in this field.
I
will follow the textbook (EMT) more or less. There is a chapter on
unbounded operators in EMT and chapter on Banach algebras as well.
Sometimes I will post additional notes on this web page. The notes
on C* algebras will be based on various books. Here are some
additional references:
Introductory
Functional Analysis with Applications, by Erwin Kreyszig,
Wiley. (fairly elementary)
Essential Results of
Functional Analysis, by Robert J. Zimmer, The University of
Chicago Press (Short bu sophisticated)
Functional Analysis, by
Reed and Simon, Academic Press (fairly comprehensive with
lots of interesting notes)
Fourier Analysis and
Self-adjointness, Reed and Simon, Academic Press (lots of
mathematics is in this volume)
Perturbation theory of
linear operators, by Tosio Kato, Springer (one of the standard
works on functional analysis)
Linear Operators in
Hilbert Space, by Joachim Weidmann, Springer (very clear
writing)
Theory of linear operators
in Hilbert Space, by Akhiezer and Glazman (a classic)
An invitation to C*
algebras, by William Arveson (a standard work in this field)
Normed Rings by M.A.
Naimark ( a classic with an enormous amount of math in it)
Here is a rough outline of the course with an
approximate number of lectures devoted to each topic:
Review of
Hilbert space theory and linear operators: 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. Bounded operators,
completely continuous operators, projection operators,
unitary operators with examples such as the
Fourier transform. This is mainly to familiarize students with
these notions who did not take Math 7338. 1-2 weeks
General concepts
for linear (unbounded) operators: Closed operators,
invariant subspaces, resolvent and spectrum, range and kernel of
unbounded
operators,adjoint operators,
symmetric and selfadjoint operators, fundamental criterion for
self-adjointness, Cayley transform, spectral theorem for undbounded
self-adjoint
with examples mainly differential operators and
singular integrals. 2-3 weeks
Extension of operators:
symmetric operators versus selfadjoint operators, deficiency
indices, self-adjoint extensions, semi-bounded operators,
Kato-Rellich theorem, examples 2 weeks
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. 1 week
Banach Algebras: Ideals,
maximal ideals, Gelfand map, Gelfand Naimark theorem. 2 weeks
Introduction to C* algebras:
States, GNS construction, commutative C* algebras, von Neumann
algebras, spectrum of C* algebras, 3-4 weeks
Notes on unbounded
operators
Homework
GRADES: There will be no tests but homework, which
you are required to hand in. The grade will be awarded according to
correctly solved
homework problems. The grading scheme is A for Homework score 90%
and above, B homework scores in (90, 80]%,
C homework scores in (80,70]%, D homework scores
in (70,60]% and F below 60%.