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A densely defined operator equal to its adjoint, including equality of domains.
Symmetric is weaker than self-adjoint for unbounded operators.
wp-self_adjoint_operatornlab-self_adjoint_operatorEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Self-adjointness requires equality with the adjoint including the operator domain.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-self_adjoint_operatornlab-self_adjoint_operatorEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
A possibly unbounded self-adjoint operator is closed and densely defined, so its resolvent and spectrum are well-defined; its spectrum is real.
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.
wp-self_adjoint_operatornlab-self_adjoint_operatorwp-operator_spectrumnlab-operator_spectrumThis is an authored directed relation from the source endpoint to the target endpoint.
A positive operator is self-adjoint with nonnegative expectation values.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-self_adjoint_operatornlab-self_adjoint_operatorwp-positive_operatornlab-positive_operatortong-quantumThis is an authored directed relation from the source endpoint to the target endpoint.
Sharp quantum observables are represented by self-adjoint operators or their spectral measures.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
wp-physics-observablenlab-physics-quantum_observablewp-self_adjoint_operatornlab-self_adjoint_operatorThis is an authored directed relation from the source endpoint to the target endpoint.
The spectral theorem associates a projection-valued measure to a self-adjoint operator.
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.
wp-spectral_measurenlab-spectral_measureThis is an authored directed relation from the source endpoint to the target endpoint.
The spectral theorem gives a projection-valued measure and functional calculus.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-self_adjoint_operatornlab-self_adjoint_operatorwp-spectral_theoremnlab-spectral_theoremtong-quantum