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

Fredholm Alternative

Homework 3, Due April 4                                                                              Eigenvalues for Compact Operators

Homework 4, due April 24                                                                             Basic theorem on self-adjointness

Notes on isometries