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
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%.