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A linear map from a vector space to itself, or more generally between vector spaces.
In analysis, “operator” often carries additional domain, topology, or boundedness data.
wp-linear_operatornlab-linear_operatorEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
A linear operator is additional data: a chosen linear map from a vector space to itself, or more generally between specified vector spaces.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-linear_operatornlab-linear_operatorEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Boundedness is equivalent to continuity for linear maps between normed spaces.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-bounded_linear_operatornlab-bounded_linear_operatorThis is an authored directed relation from the source endpoint to the target endpoint.
A densely defined operator may be unbounded and is specified on a dense subspace.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-densely_defined_operatornlab-densely_defined_operatorThis is an authored directed relation from the source endpoint to the target endpoint.
For an operator and specified domain or boundary conditions, a Green function is a fundamental-solution or inverse kernel when such a distributional inverse exists.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-greens_functionnlab-greens_functionThis is an authored directed relation from the source endpoint to the target endpoint.
Whenever both products are defined, operator composition defines the anticommutator .
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-operator-anticommutatornlab-operator-anticommutatorThis is an authored directed relation from the source endpoint to the target endpoint.
Whenever both products are defined on a suitable common domain, operator composition defines the commutator .
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-operator-commutatornlab-operator-commutatorThis is an authored directed relation from the source endpoint to the target endpoint.
An algebraic projection is an everywhere-defined linear operator satisfying ; on a Hilbert space, an orthogonal projection additionally satisfies and is bounded.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-projection_operatornlab-projection_operator