Spectral Theory

Spectral theory generalizes the eigenvalues of a matrix to operators on infinite-dimensional spaces, describing an operator through its spectrum and, for self-adjoint operators, a spectral decomposition.

Definition

Spectral theory studies the spectrum of a linear operator, the set of scalars for which the operator minus that scalar fails to be invertible, and represents suitable operators, especially self-adjoint ones, in terms of that spectrum through a spectral measure.

Scope

This topic covers the spectrum, resolvent set, and resolvent of a bounded operator, the partition of the spectrum into point, continuous, and residual parts, the spectral radius formula, the spectral theorem for compact self-adjoint operators with its eigenfunction expansion, and the spectral theorem for general bounded self-adjoint and normal operators via projection-valued measures and the functional calculus.

Core questions

• How is the spectrum defined, and how does it extend the notion of eigenvalues?
• What is the structure of the spectrum of a compact self-adjoint operator?
• How does the spectral theorem represent a self-adjoint operator?
• What is the functional calculus, and how does it let functions act on operators?

Key theories

Spectral theorem for compact self-adjoint operators

A compact self-adjoint operator has an orthonormal basis of eigenvectors with real eigenvalues accumulating only at zero, giving a diagonalization that directly generalizes the finite-dimensional case.

Spectral theorem and functional calculus

Every bounded self-adjoint, and more generally normal, operator is represented as an integral against a projection-valued spectral measure, allowing bounded functions of the operator to be defined and manipulated.

Clinical relevance

Spectral theory is the mathematical core of quantum mechanics, where the spectrum of a self-adjoint operator gives the possible measured values of an observable; it also underlies vibration and stability analysis, the eigenfunction methods for partial differential equations, and spectral techniques in data analysis and graph theory.

History

Hilbert introduced the term spectrum in his study of integral equations, and the theory of self-adjoint operators was completed by von Neumann in the late 1920s, who established the spectral theorem for unbounded operators to provide rigorous foundations for quantum mechanics.

Key figures

• David Hilbert
• John von Neumann
• Frigyes Riesz

Seminal works

• conway1985
• reedsimon1980

Frequently asked questions

What is the spectrum of an operator?

It is the set of scalars for which the operator minus that scalar multiple of the identity is not invertible; for matrices this is exactly the set of eigenvalues, but in infinite dimensions it can also include non-eigenvalue points.

Why is the spectral theorem so important?

It diagonalizes self-adjoint operators, just as symmetric matrices are diagonalized, which is what makes self-adjoint operators the natural model for physical observables and enables functions of operators to be defined.