Nuclear operators between Banach spaces: Difference between revisions

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In [[mathematics]], a '''nuclear operatoroperators between Banach spaces''' isare a [[compactlinear operator]]s forbetween which[[Banach aspace]]s in infinite dimensions that share some of the properties of their counter-part in finite dimension. In [[traceHilbert (linear algebra)|tracespace]]s maysuch beoperators defined,are suchusually thatcalled the[[trace class|trace isclass finiteoperators]] and independentone ofcan thedefine choicesuch ofthings basisas (atthe least[[trace on(linear wellalgebra)|trace]]. behavedIn Banach spaces; therethis areis someno spaceslonger onpossible whichfor general nuclear operators, doit notis havehowever apossible for <math>\tfrac{2}{3}</math>-nuclear operator via the [[Grothendieck trace) theorem]].

Nuclear operators are essentially the same as '''[[trace class|trace-class operators]]''', though most authors reserve the term "trace-class operator" for the special case of nuclear operators on [[Hilbert space]]s.

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; for more details about the important special case of nuclear (= trace-class) operators on Hilbert space, see the article [[Trace class]].

== Compact operator ==

== Nuclear operators on Hilbert spaces ==

{{main|trace class operator}}

An operator <math>\mathcal L</math> on a [[Hilbert space]] <math>\mathcal H</math>

<math display="block">\mathcal{L} : \mathcal{H} \to \mathcal{H}</math>

is [[compact operator|compact]] if it can be written in the form{{Citation needed|date=September 2011}}

<math display="block">\mathcal{L} = \sum_{n=1}^N \rho_n \langle f_n, \cdot \rangle g_n,</math>

where <math>1 \leq N \leq \infty,</math> and <math>\{f_1, \ldots, f_N\}</math> and <math>\{g_1, \ldots, g_N\}</math> are (not necessarily complete) orthonormal sets. Here <math>\{\rho_1, \ldots, \rho_N\}</math> areis a set of real numbers, the set of [[singular value]]s of the operator, obeying <math>\rho_n \to 0</math> if <math>N = \infty.</math>

The bracket <math>\langle\cdot, \cdot\rangle</math> is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm.

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<math display="block">\sum_{n=1}^\infty |\rho_n| < \infty.</math>

=== Properties ===

A nuclear operator on a Hilbert space has the important property that a [[Trace class|trace]] operation may be defined. Given an orthonormal basis <math>\{\psi_n\}</math> for the Hilbert space, the trace is defined as

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Obviously, the sum converges absolutely, and it can be proven that the result is independent of the basis{{Citation needed|date=September 2011}}. It can be shown that this trace is identical to the sum of the eigenvalues of <math>\mathcal{L}</math> (counted with multiplicity).

== OnNuclear operators on Banach spaces ==

{{Main|Fredholm kernel}}

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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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== Examples ==

Suppose that <math>f : H_1 \to H_2</math> and <math>g : H_2 \to H_3</math> are [[Hilbert-Schmidt operator]]s between Hilbert spaces. Then the composition <math>g \circ f : H_1 \to H_2H_3</math> is a [[nuclear operator]].{{sfn|Schaefer|Wolff|1999|p=177}}

== See also ==

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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]]