Calkin algebra (original) (raw)
In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. Here the addition in B(H) is addition of operators and the multiplication in B(H) is composition of operators; it is easy to verify that these operations make B(H) into a ring. When scalar multiplication is also included, B(H) becomes in fact an algebra over the same field over which H is a Hilbert space.
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| dbo:abstract | In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. Here the addition in B(H) is addition of operators and the multiplication in B(H) is composition of operators; it is easy to verify that these operations make B(H) into a ring. When scalar multiplication is also included, B(H) becomes in fact an algebra over the same field over which H is a Hilbert space. (en) In der Mathematik ist die Calkin-Algebra (nach John Williams Calkin) eine spezielle Banachalgebra, die einem Banachraum zugeordnet ist. In der Calkin-Algebra kann man Eigenschaften stetiger linearer Operatoren vereinfacht betrachten, indem Operatoren, deren Differenz kompakt ist, identifiziert werden. So kommt man zu Klassifikationssätzen für normale Operatoren modulo kompakter Operatoren. (de) En analyse fonctionnelle — une branche des mathématiques — l'algèbre de Calkin d'un espace de Banach E est le quotient de l'algèbre de Banach B(E) des opérateurs bornés sur E par l'idéal fermé K(E) des opérateurs compacts. C'est donc encore une algèbre de Banach, pour la norme quotient. Lorsque l'espace E n'est pas précisé, il s'agit de l'espace de Hilbert H séparable et de dimension infinie. Son algèbre de Calkin permet de classifier entre autres les opérateurs normaux sur H, modulo les opérateurs compacts. (fr) |
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| rdfs:comment | In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. Here the addition in B(H) is addition of operators and the multiplication in B(H) is composition of operators; it is easy to verify that these operations make B(H) into a ring. When scalar multiplication is also included, B(H) becomes in fact an algebra over the same field over which H is a Hilbert space. (en) In der Mathematik ist die Calkin-Algebra (nach John Williams Calkin) eine spezielle Banachalgebra, die einem Banachraum zugeordnet ist. In der Calkin-Algebra kann man Eigenschaften stetiger linearer Operatoren vereinfacht betrachten, indem Operatoren, deren Differenz kompakt ist, identifiziert werden. So kommt man zu Klassifikationssätzen für normale Operatoren modulo kompakter Operatoren. (de) En analyse fonctionnelle — une branche des mathématiques — l'algèbre de Calkin d'un espace de Banach E est le quotient de l'algèbre de Banach B(E) des opérateurs bornés sur E par l'idéal fermé K(E) des opérateurs compacts. C'est donc encore une algèbre de Banach, pour la norme quotient. Lorsque l'espace E n'est pas précisé, il s'agit de l'espace de Hilbert H séparable et de dimension infinie. Son algèbre de Calkin permet de classifier entre autres les opérateurs normaux sur H, modulo les opérateurs compacts. (fr) |
| rdfs:label | Calkin-Algebra (de) Calkin algebra (en) Algèbre de Calkin (fr) |
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