Spectral theory

Spectral Theory

Spectral theory has for its origin the classical theory of finite matrices and, in a sense, includes all theories which may be specialized to some particular phase of that classical theory. Such a meaning for the term spectral theory is perhaps too broad, for its acceptance would force one, as is becoming increasingly evident, to admit that spectral theory embraces not a small part of such fiel...

Spectral Theory

Spectral Theory

In many applications it is important to understand the spectral properties of a linear operator T : X → X, where X is some vector space over IR or I C. In the finite dimensional (complex) case linear operators may be characterised as matrices and the Jordan normal form theorem applies, providing a basis of generalised eigenvectors. If, in addition, T is normal (i.e. T and T ∗ commute) with resp...

Spectral Theory

These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensio...

Phenomena in Spectral Theory

The extremal problem for the functional determinant of a natural linear elliptic operator a on Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, we encounter non-linear geometric analytic phenomena: sharp inequalities comparing nonlinear functionals of the metric and its derivatives. The derivation and use of such inequalities in new situations, esp...