Nuclear operators between Banach spaces: Difference between revisions

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In [[mathematics]], a '''nuclear operatoroperators between Banach spaces''' isare a [[linear operator]]s between [[Banach space]]s in infinite dimensions that share some of the properties of their counter-part in finite dimension. In [[Hilbert space]]]s such operators are usually called [[trace class|trace class operators]] and one can define such things as the [[trace (linear algebra)|trace]]. In Banach spaces this is no longer possible for general nuclear operators, it is however possible for <math>\tfrac{2}{3}</math>-nuclear operator via the [[Grothendieck trace theorem]].

The general definition for [[Banach space]]s was given by [[Grothendieck]]. This article presents both cases but concentrates on the general case of nuclear operators on Banach spaces.

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=== Relation to trace-class operators ===

With additional steps, a trace may be defined for such operators when <math>A = B.</math>

=== Properties ===

The trace and determinant can no longer be defined in general in Banach spaces. However they can be defined for the so-called <math>\tfrac{2}{3}</math>-nuclear operators via [[Grothendieck trace theorem]].

=== Generalizations ===

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* {{Citation |first1=Stephan |last1=Stolz |first2=Peter |last2=Teichner |title=Traces in monoidal categories |journal=Transactions of the American Mathematical Society |volume=364 |year=2012 |issue=8 |pages=4425–4464 |mr=2912459 |doi=10.1090/S0002-9947-2012-05615-7 |arxiv=1010.4527}}

{{Functional Analysisanalysis}}

{{Topological tensor products and nuclear spaces}}

{{TopologicalTensorProductsAndNuclearSpaces}}

[[Category:Operator theory]]

[[Category:Topological tensor products]]

[[Category:Linear operators]]