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Canonical static concept record

Adjoint operator

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Summary

For a densely defined operator between Hilbert spaces, the operator determined on its natural domain by $\langle Ax,y\rangle=\langle x,A^*y\rangle$ .

Record metadata

Canonical name: Adjoint operator
Concept ID: adjoint_operator
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: Operator construction

Carrier(s)

Hilbert spaces
densely defined operator $A$

Data

operator $A^*$ on its natural domain

Axioms / constraints

$\langle Ax,y\rangle=\langle x,A^*y\rangle$

Canonically induces

symmetric and self-adjoint operator notions

Notes

For unbounded operators, equality of domains is essential and the adjoint is always closed.

Concept sources

Wikipedia — Hermitian adjoint — Hermitian adjoint · encyclopedic reference · source ID wp-adjoint_operator
nLab — adjoint operator — adjoint operator · mathematical reference · source ID nlab-adjoint_operator

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Densely defined operator Adjoint operator

Permalink to relation

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

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

Authored explanation

An inner product determines the adjoint and its domain.

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 — Hermitian adjoint — Hermitian adjoint · encyclopedic reference · source ID wp-adjoint_operator
nLab — adjoint operator — adjoint operator · mathematical reference · source ID nlab-adjoint_operator

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.