Seminar in Functional Analysis: Operator Theory

Math 7380
Louisiana State University
Fall Semester, 2025

Prof. Stephen Shipman

Place: Lockett 119
Time: 2:30-3:20 Monday, Wednesday, Friday

Office: Room 314 of Lockett Hall
Email: shipman@lsu.edu
Office Hours: Monday 1:00-2:20, Wednesday and Friday 3:30-5:00, and by appointment.

Course Description

The material in this course is foundational broadly in analysis and its applications. Linear operator theory is ubiquitous in all flavors of mathematical physics, including PDE, quantum mechanics, and quantum field theory. The topics include compact operators, index theory, and spectral theory, and Banach algebras.

The course will approximately follow the AMS book Operator Theory by Barry Simon.

• Chapter 1: Linear algebra in finite dimension and functional analysis basics
  
  • Normal forms
  • Eigenvalues of a family A(z)
  • Perturbation of eigenvalues
  • Tensor products
  • Basic theorems from functional analysis
• Chapter 2: Operator basics
  
  • Unbounded operators
  • Transpose and adjoint
  • Projections
  • Self-adjoint, normal, and unitary operators
  • Spectrum, basic spectral theorem
  • Analytic functional calculus
  • Polar decomposition
• Chapter 3: Compact and Fredholm operators
  
  • Compact, completely continuous, and finite-approximable operators
  • In Banach space and in Hilbert space
  • Structure of compact operators: Schauder and Ringrose theorems
  • Fredholm operators: index theory, examples, and applications
• Chapter 5: The spectral theorem
  
  • Resolution of a Hilbert space
  • Functional calculus
  • Spectral theorem
  • Rank-one perturbations
  • Major operator theorems concerning spectrum
• Chapter 6: Banach algebras
  
  • Banach algebras
  • C^* algebras
  • Gelfand transform
  • Abstract spectral theorem
• Unbounded self-adjoint operators
  
  • Closed unbounded operators
  • Domains of unbounded operators
  • Self-adjoint unbounded operators
  • Form domains
  • Schrödinger operators
  • Spectral theorem
  • Self-adjoint extensions of symmetric operators
  • Rayleigh quotient

Prerequisite

Assignments and Evaluation

I will give you problem sets throughout the semester. Of course, to really learn the material, you should do as many as you can. However the grade in the course will be based on just a few problems (maybe one to three of your choice) per set. There will no mid-term or final exams.

This is my general policy on academic integrity.

Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.

Problem Sets

Grading scale (required by LSU)

A+: at least 95%	A: at least 90%	A-: at least 88%
B+: at least 85%	B: at least 80%	B-: at least 78%
C+: at least 75%	C: at least 70%	C-: at least 68%
D+: at least 65%	D: at least 60%	D-: at least 50%
F: less than 50%

Bibliography

1. Excerpts on compact operators:
   Naylor and Sell
   Kress
   Reed and Simon
   Riesz and Sz.-Nagy
   Akhiezer and Glazman, Chapter V; see the link below.
2. Michael Reed and Barry Simon, Methods of Modern Mathematics: Vol. I Functional Analysis, Vol. II Fourier Analysis and Self-Adjointness, Vol. III Scattering Theory, Vol. IV Analysis of Operators, excerpts from Vol. I (spectral theorem), Vol. II (self-adjoint extensions), Academic Press, 1980.
3. Jean-Pierre Aubin, Applied Functional Analysis, Wiley, 2000.
4. Hans L. Cycon and Richard G. Froese and Werner Kirsch and Barry Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, 1987.
5. Werner Kirsch, An Invitation to Random Schrödinger operators, Notes, 2007.
6. I. Gohberg and P. Lancaster and L. Rodman, Indefinite Linear Alegebra and Applications, Birkhäuser, 2005.
7. Gerald B. Folland, Introduction to Partial Differential Equations, Chapter 0, Second Edition, Princeton University Press, 1995.
8. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.