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Canonical static concept record

Positive operator

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Summary

A self-adjoint operator $A$ satisfying $\langle\psi,A\psi\rangle\ge0$ on its domain.

Record metadata

Canonical name: Positive operator
Concept ID: positive_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 class

Carrier(s)

Hilbert space $H$

Data

bounded or suitably defined self-adjoint operator $A$

Axioms / constraints

$\langle x,Ax\rangle\ge0$ for all vectors in the domain

Canonically induces

positive square root
order structure on self-adjoint operators

Notes

For unbounded positive operators, domain and self-adjointness conditions must be stated.

Concept sources

Wikipedia — Positive operator — Positive operator · encyclopedic reference · source ID wp-positive_operator
nLab — positive operator — positive operator · mathematical reference · source ID nlab-positive_operator
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 Positive operator

Permalink to relation

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

Relation ID: positive_from_selfadjoint
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: nonnegative quadratic form

Authored explanation

A positive operator is self-adjoint with nonnegative expectation values.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

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 — Positive operator — Positive operator · encyclopedic reference · source ID wp-positive_operator
nLab — positive operator — positive operator · mathematical reference · source ID nlab-positive_operator
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.