Spectral theory
MATH-561 / 5 credits
Teacher: Ventura Paolo
Language: English
Remark: Cours donné en alternance tous les deux ans.
Summary
This course is an introduction to the spectral theory of linear operators acting in Hilbert spaces. The main goal is the spectral decomposition of unbounded selfadjoint operators. We will also give elementary applications to the analysis of PDEs.
Content
The first chapter will recall basic properties of Hilbert spaces, linear operators and their spectra.
We will then introduce the notion of integral with respect to a spectral family, which leads naturally to the spectral decomposition of symmetric operators, i.e. bounded selfadjoint operators. This result, based on the Riemann-Stieltjes integration theory, will be our first spectral theorem.
The third chapter will present an approximation procedure which allows one to extend the spectral theorem to unbounded selfadjoint operators, using the Lebesgue-Stieltjes integral. Our approach follows closely the original proof given by Frigyes Riesz in 1952. (If time permits, we will also discuss an alternative proof due to John von Neumann.)
Finally, as an important consequence of the spectral theorem, we will prove Stone's theorem describing the structure of one-parameter unitary groups. We will conclude the course by introducing the Lumer-€“Phillips theorem, which provides a natural framework for PDEs through energy estimates and maximal dissipativity. This will provide a bridge from the self-adjoint theory developed in the course to more general evolution problems, where resolvent estimates become a key tool for understanding the growth and stability of solutions.
Learning Prerequisites
Required courses
Analysis I-IV; Linear algebra; Functional analysis
Learning Outcomes
By the end of the course, the student must be able to:
- Prove properties of bounded and unbounded linear operators in Hilbert spaces
- Solve problems involving symmetric / selfadjoint operators
- Demonstrate a thorough understanding of the spectral decomposition theorem and of Stone's theorem
- Explain how the theory provides a rigorous framework for the analysis of linear evolution equations arising in PDEs
Teaching methods
Blackboard lectures + exercise sessions in class
Assessment methods
Oral exam.
In case art. 3 al. 5 of the Règlement de section applies, the teacher communicates the form of the exam to the concerned students.
In the programs
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Spectral theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Spectral theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Spectral theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Spectral theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional