Fredholm operator
E391903
A Fredholm operator is a bounded linear operator between Banach (or Hilbert) spaces with finite-dimensional kernel and cokernel and a closed range, characterized by its integer-valued index.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Fredholm operator canonical | 2 |
| Fredholm index | 1 |
| Fredholm operators | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3821386 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Fredholm operator Context triple: [Atiyah–Singer index theorem, usesConcept, Fredholm operator]
-
A.
Hilbert–Schmidt operators
Hilbert–Schmidt operators are a class of compact operators on Hilbert spaces characterized by having finite Hilbert–Schmidt norm, playing a central role in functional analysis and operator theory.
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B.
Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
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C.
Jacobi operator
The Jacobi operator is a linear differential operator central to the theory of elliptic functions and integrable systems, named after the mathematician Carl Gustav Jacob Jacobi.
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D.
Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
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E.
Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Fredholm operator Target entity description: A Fredholm operator is a bounded linear operator between Banach (or Hilbert) spaces with finite-dimensional kernel and cokernel and a closed range, characterized by its integer-valued index.
-
A.
Hilbert–Schmidt operators
Hilbert–Schmidt operators are a class of compact operators on Hilbert spaces characterized by having finite Hilbert–Schmidt norm, playing a central role in functional analysis and operator theory.
-
B.
Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
-
C.
Jacobi operator
The Jacobi operator is a linear differential operator central to the theory of elliptic functions and integrable systems, named after the mathematician Carl Gustav Jacob Jacobi.
-
D.
Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
-
E.
Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
operator theory concept ⓘ |
| appearsIn |
Banach spaces
ⓘ
surface form:
Banach space theory
Hilbert space theory ⓘ |
| characterizedBy |
Fredholm operator
self-linksurface differs
ⓘ
surface form:
Fredholm index
|
| codomain |
Banach space
ⓘ
Hilbert space ⓘ |
| definition |
A bounded linear operator between Banach spaces with finite-dimensional kernel and cokernel and closed range.
ⓘ
A bounded linear operator between Hilbert spaces with finite-dimensional kernel and cokernel and closed range. ⓘ |
| domain |
Banach space
ⓘ
Hilbert space ⓘ |
| field |
functional analysis
ⓘ
operator theory ⓘ |
| generalizes | Fredholm integral operator ⓘ |
| hasDualConcept | adjoint Fredholm operator ⓘ |
| hasInvariant |
index(T) = dim ker(T) − codim ran(T)
ⓘ
index(T*) = − index(T) ⓘ stable index under compact perturbations ⓘ |
| hasProperty |
Fredholm index is locally constant on space of Fredholm operators
ⓘ
Fredholm operators form an open subset in the space of bounded operators ⓘ Fredholmness is preserved under compact perturbations ⓘ Fredholmness is preserved under composition with invertible operators ⓘ Fredholmness is preserved under taking adjoints ⓘ adjoint of a Fredholm operator is Fredholm ⓘ bounded linear operator ⓘ closed range ⓘ cokernel is finite-dimensional ⓘ finite-dimensional cokernel ⓘ finite-dimensional kernel ⓘ invertible operator has index 0 ⓘ kernel is finite-dimensional ⓘ range has finite codimension ⓘ |
| hasSpecialCase | invertible operator ⓘ |
| indexType | integer-valued index ⓘ |
| namedAfter | Ivar Fredholm ⓘ |
| relatedTo |
Atkinson theorem
ⓘ
C*-algebras ⓘ Fredholm alternative ⓘ K-theory ⓘ elliptic differential operator ⓘ index theory ⓘ spectral theory ⓘ |
| stableUnder |
compact perturbations
ⓘ
small norm perturbations preserving Fredholmness ⓘ |
| usedIn |
formulation of Atiyah–Singer index theorem
ⓘ
noncommutative geometry ⓘ study of partial differential equations ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fredholm operator Description of subject: A Fredholm operator is a bounded linear operator between Banach (or Hilbert) spaces with finite-dimensional kernel and cokernel and a closed range, characterized by its integer-valued index.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Functional analysis
this entity surface form:
Fredholm operators
this entity surface form:
Fredholm index