Riesz projection
E747348
The Riesz projection is a linear operator in functional analysis that projects onto the invariant subspace associated with a portion of the spectrum of a bounded linear operator, defined via contour integration of its resolvent.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in functional analysis
ⓘ
linear operator ⓘ projection operator ⓘ |
| actsOn |
Banach space
ⓘ
Hilbert space ⓘ |
| appliesTo | bounded linear operators on Banach spaces ⓘ |
| associatedWith |
bounded linear operator
ⓘ
spectrum of an operator ⓘ |
| belongsTo | holomorphic functional calculus ⓘ |
| belongsToTheory | spectral theory of linear operators ⓘ |
| canBeExtendedTo | certain unbounded closed operators ⓘ |
| commutesWith | the operator T NERFINISHED ⓘ |
| constructedUsing | resolvent (zI - T)^{-1} ⓘ |
| correspondsTo | portion of the spectrum enclosed by Γ ⓘ |
| definedBy | contour integral of the resolvent ⓘ |
| dependsOn | choice of contour Γ ⓘ |
| generalizes | spectral projections for normal operators ⓘ |
| hasAlternativeName | spectral projection in the sense of Riesz ⓘ |
| hasComplement | I - P as projection onto complementary invariant subspace ⓘ |
| hasDefinition | P = (1/(2πi)) ∮_Γ (zI - T)^{-1} dz ⓘ |
| hasKernel | invariant subspace for T corresponding to complement of selected spectrum ⓘ |
| hasNorm | at least 1 unless trivial ⓘ |
| hasProperty | range equals direct sum of generalized eigenspaces for eigenvalues inside Γ (in finite dimensions) ⓘ |
| hasRange | invariant subspace for T ⓘ |
| isBounded | true ⓘ |
| isIdempotent | true ⓘ |
| isLinear | true ⓘ |
| isOrthogonalProjection |
not necessarily
ⓘ
yes if T is normal on a Hilbert space and Γ selects part of spectrum ⓘ |
| isSpectralProjection | true ⓘ |
| isWellDefinedIf |
spectrum inside Γ is isolated from rest of spectrum
ⓘ
Γ lies in the resolvent set of T ⓘ |
| namedAfter | Frigyes Riesz NERFINISHED ⓘ |
| projectsOnto |
invariant subspace of an operator
ⓘ
spectral subspace ⓘ |
| requires |
closed contour Γ in the resolvent set of T
ⓘ
resolvent to be analytic on and inside Γ ⓘ |
| satisfies |
PT = TP
ⓘ
P^2 = P ⓘ spectrum of T|_{ker(P)} lies in complement of selected part of spectrum ⓘ spectrum of T|_{range(P)} lies in selected part of spectrum ⓘ |
| usedFor |
decomposing a space into spectral subspaces
ⓘ
isolating eigenvalues and their generalized eigenspaces ⓘ spectral decomposition of bounded operators ⓘ |
| usedIn |
Fredholm theory
NERFINISHED
ⓘ
perturbation theory of linear operators ⓘ study of isolated eigenvalues of finite multiplicity ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.