Applied linear operators and spectral methods

Welcome to this learning project about Applied linear operators and spectral methods!

• Project code:
• Suggested Prerequisites:
  • Introduction to real analysis
  • Introduction to ordinary differential equations
• Time investment: 6 months
• Portal:Mathematics
• School:Applied Mathematics
• Level: Graduate

The course covers the theory of linear operators applied to matrix, differential, and integral equations. We also discuss Fredholdm integral equations and spectral theory.

This learning project aims to help applied mathematicians and engineers gain the understanding needed to read and critically assess papers that deal with functional analysis in its various forms.

We will talk about operator theory and linear operators as applied to functional analysis.

• Principles of Applied Mathematics, Transformation and Approximation: 2nd Edition, by J. Keener, 2000.
• Green's Functions and Boundary Value Problems, by Ivar Stakgold, Wiley.
• Methods of Modern Mathematical Physics. I - Functional Analysis, by Michael Read and Barry Simon, Elsevier.
• Applied Functional Analysis: Applications to Mathematical Physics, by Eberhard Zeidler, Springer Series on the Applied Mathematical Sciences, vol. 108. This is a more technical book than the others..
• Applied Functional Analysis: Main Principles and their Applications, by Eberhard Zeidler, Springer Series on the Applied Mathematical Sciences, vol. 109. This is a more technical book than the others..

• Lecture 1: Linear vector spaces
• Lecture 2: Normed inner product spaces, orthogonal vectors, linear operators
• Lecture 3: Adjoint operators, spectral theory for matrices
• Lecture 4: More on spectral decompositions
• Differentiation of distributions
• Convergence of distributions
• Differential equations in the sense of distributions
• Differentiation of distributions with severe discontinuities
• Green's functions for linear differential operators

• Assignment 1: ...

Subject classification: this is a mathematics resource.

Educational level: this is a tertiary (university) resource.

Completion status: About halfway there. You may help to clarify and expand it.