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Canonical static concept record

Spectral theorem for self-adjoint operators

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Summary

A self-adjoint operator admits a projection-valued spectral resolution and functional calculus.

Record metadata

Canonical name: Spectral theorem for self-adjoint operators
Concept ID: spectral_theorem
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Theorem

Carrier(s)

self-adjoint operator $A$ on a complex Hilbert space $H$

Data

projection-valued spectral measure or multiplication-operator representation

Axioms / constraints

hypotheses depend on bounded versus unbounded case

Canonically induces

functional calculus
spectral decomposition

Notes

There are distinct finite-dimensional, bounded-normal, compact, and unbounded-self-adjoint versions.

Concept sources

Wikipedia — Spectral theorem — Spectral theorem · encyclopedic reference · source ID wp-spectral_theorem
nLab — spectral theorem — spectral theorem · mathematical reference · source ID nlab-spectral_theorem
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Self-adjoint operator Spectral theorem for self-adjoint operators

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: spectral_from_selfadjoint
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: spectral resolution

Authored explanation

The spectral theorem gives a projection-valued measure and functional calculus.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Wikipedia — Self-adjoint operator — Self-adjoint operator · encyclopedic reference · source ID wp-self_adjoint_operator
nLab — self-adjoint operator — self-adjoint operator · mathematical reference · source ID nlab-self_adjoint_operator
Wikipedia — Spectral theorem — Spectral theorem · encyclopedic reference · source ID wp-spectral_theorem
nLab — spectral theorem — spectral theorem · mathematical reference · source ID nlab-spectral_theorem
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

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Each relation below starts at this concept.