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Canonical static concept record

Operator spectrum

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Summary

For a bounded operator on a complex Banach space, or a closed densely defined operator, the scalars $\lambda$ for which $A-\lambda I$ has no everywhere-defined bounded inverse.

Record metadata

Canonical name: Operator spectrum
Concept ID: operator_spectrum
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: Spectral invariant

Carrier(s)

complex Banach space and bounded operator, or closed densely defined operator

Data

set $\sigma(A)$ of spectral parameters

Axioms / constraints

$\lambda\in\sigma(A)$ iff $A-\lambda I$ lacks an everywhere-defined bounded inverse

Canonically induces

point, continuous, and residual spectral parts under appropriate definitions

Notes

For unbounded operators the domain and closedness are essential.

Concept sources

Wikipedia — Spectrum (functional analysis) — Spectrum (functional analysis) · encyclopedic reference · source ID wp-operator_spectrum
nLab — spectrum of an operator — spectrum of an operator · mathematical reference · source ID nlab-operator_spectrum

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Bounded linear operator Operator spectrum

Permalink to relation

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

Relation ID: math_linear_operator_to_operator_spectrum
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: spectrum

Authored explanation

A bounded operator on a complex Banach space determines its spectrum by failure of $A-\lambda I$ to have an everywhere-defined bounded inverse.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Spectrum (functional analysis) — Spectrum (functional analysis) · encyclopedic reference · source ID wp-operator_spectrum
nLab — spectrum of an operator — spectrum of an operator · mathematical reference · source ID nlab-operator_spectrum

Self-adjoint operator Operator spectrum

Permalink to relation

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

Relation ID: self_adjoint_to_operator_spectrum
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: spectrum of a self-adjoint operator

Authored explanation

A possibly unbounded self-adjoint operator is closed and densely defined, so its resolvent and spectrum are well-defined; its spectrum is real.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

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 — Spectrum (functional analysis) — Spectrum (functional analysis) · encyclopedic reference · source ID wp-operator_spectrum
nLab — spectrum of an operator — spectrum of an operator · mathematical reference · source ID nlab-operator_spectrum

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.