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 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 on [[trace class|trace class operator]]s.

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 on [[trace class|trace class operator]]s.

==Compact operator==

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

== Nuclear operators on Hilbert spaces ==

:<math>\mathcal{L}:\mathcal{H} \to \mathcal{H}</math>

{{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 and only 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 \leleq N \leleq \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.

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.

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

An operator that is compact as defined above is said to be '''{{em|nuclear'''}} or '''{{em|trace-class'''}} if

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

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

=== Properties ===

where <math>1 \le N \le \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> are a set of real numbers, the [[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.

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

An operator that is compact as defined above is said to be '''nuclear''' or '''trace-class''' if

:<math display="block">\mboxoperatorname{Tr} \mathcal {L} = \sum_n \langle \psi_n , \mathcal{L} \psi_n \rangle.</math>.

It is immediateObviously, 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).

:<math>\sum_{n=1}^\infty |\rho_n| < \infty</math>

==On Nuclear operators on Banach spaces ==

==Properties==

{{Main|Fredholm kernel}}

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

The definition of trace-class operator was extended to [[Banach space]]s by [[Alexander Grothendieck]] in 1955.

:<math>\mbox{Tr} \mathcal {L} = \sum_n \langle \psi_n , \mathcal{L} \psi_n \rangle</math>.

Let ''<math>A''</math> and ''<math>B''</math> be Banach spaces, and ''<math>A'''^{\prime}</math> be the [[continuousContinuous dual space|dual]] of ''<math>A'',</math> that is, the set of all [[continuousContinuous (topology)|continuous]] or (equivalently) [[bounded linear functional]]s on ''<math>A''</math> with the usual norm. Then an operator

It is immediate 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).

There is a canonical evaluation map

<math display="block">A^{\prime} \otimes B \to \operatorname{Hom}(A, B)</math>

(from the [[projective tensor product]] of <math>A</math> and <math>B</math> to the Banach space of continuous linear maps from <math>A</math> to <math>B</math>).

It is determined by sending <math>f \in A^{\prime}</math> and <math>b \in B</math> to the linear map <math>a \mapsto f(a) \cdot b.</math>

An operator <math>\mathcal L \in \operatorname{Hom}(A,B)</math> is called {{em|nuclear}} if it is in the image of this evaluation map.<ref>{{harvtxt|Schaefer|Wolff|1999|loc=Chapter III, §7}}</ref>

=== {{mvar|q}}-nuclear operators ===

==On Banach spaces==

:''See main article [[Fredholm kernel]].''

==CompactAn operator==

The definition of trace-class operator was extended to [[Banach space]]s by [[Alexander Grothendieck]] in 1955.

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

is said to be '''{{em|nuclear of order ''<math>q'' '''</math>}} if there exist sequences of vectors <math>\{g_n\} \in B</math> with <math>\Vert g_n \Vert \leleq 1,</math>, functionals <math>\left\{f^*_n\right\} \in A'^{\prime}</math> with <math>\Vert f^*_n \Vert \leleq 1</math> and [[complex number]]s <math>\{\rho_n\}</math> with

:<math> display="block">\sum_n |\rho_n|^q < \infty,</math>

such that the operator may be written as

:<math display="block">\mathcal{L} = \sum_n \rho_n f^*_n(\cdot) g_n</math>

with the sum converging in the operator norm.

Operators that are nuclear of order 1 are called '''{{em|nuclear operators'''}}: these are the ones for which the series &sum;''&rho;<submath>n\sum \rho_n</submath>'' is absolutely convergent. Nuclear operators of order 2 are called [[Hilbert–Schmidt operator]]s.

Let ''A'' and ''B'' be Banach spaces, and ''A''' be the [[continuous dual space|dual]] of ''A'', that is, the set of all [[continuous (topology)|continuous]] or (equivalently) [[bounded linear functional]]s on ''A'' with the usual norm. Then an operator

Nuclear operators of order 2 are called [[Hilbert–Schmidt operator]]s.

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

:<math>\mathcal{L}:A \to B</math>

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

is said to be '''nuclear of order ''q'' ''' if there exist sequences of vectors <math>\{g_n\} \in B</math> with <math>\Vert g_n \Vert \le 1</math>, functionals <math>\{f^*_n\} \in A'</math> with <math>\Vert f^*_n \Vert \le 1</math> and [[complex number]]s <math>\{\rho_n\}</math> with

=== Properties ===

:<math> \sum_n |\rho_n|^q < \infty,</math>

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 ===

such that the operator may be written as

More generally, an operator from a [[locally convex topological vector space]] ''<math>A''</math> to a Banach space ''<math>B''</math> is called '''{{em|nuclear'''}} if it satisfies the condition above with all ''f_n<supmath>f_n^*</supmath>'' bounded by 1 on some fixed neighborhood of 0.

:<math>\mathcal{L} = \sum_n \rho_n f^*_n(\cdot) g_n</math>

An extension of the concept of nuclear maps to arbitrary [[Monoidal category|monoidal categories]] is given by {{harvtxt|Stolz|Teichner|2012}}.

with the sum converging in the operator norm.

A monoidal category can be thought of as a [[Category (mathematics)|category]] equipped with a suitable notion of a tensor product.

An example of a monoidal category is the category of Banach spaces or alternatively the category of locally convex, complete, Hausdorff spaces; both equipped with the projective tensor product.

A map <math>f : A \to B</math> in a monoidal category is called {{em|thick}} if it can be written as a composition

<math display="block">A \cong I \otimes A \stackrel{t \otimes \operatorname{id}_A} \longrightarrow B \otimes C \otimes A \stackrel{\operatorname{id}_B \otimes s} \longrightarrow B \otimes I \cong B</math>

for an appropriate object <math>C</math> and maps <math>t: I \to B \otimes C, s: C \otimes A \to I,</math> where <math>I</math> is the monoidal unit.

In the monoidal category of Banach spaces, equipped with the projective tensor product, a map is thick if and only if it is nuclear.<ref>{{harvtxt|Stolz|Teichner|2012|loc=Theorem 4.26}}</ref>

== 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_3</math> is a [[nuclear operator]].{{sfn|Schaefer|Wolff|1999|p=177}}

== See also ==

With additional steps, a trace may be defined for such operators when ''A'' = ''B''.

* {{annotated link|Topological tensor product}}

Operators that are nuclear of order 1 are called '''nuclear operators''': these are the ones for which the series &sum;''&rho;_n'' is absolutely convergent. Nuclear operators of order 2 are called [[Hilbert–Schmidt operator]]s.

* {{annotated link|Nuclear operator}}

* {{annotated link|Nuclear space}}

== References ==

More generally, an operator from a [[locally convex topological vector space]] ''A'' to a Banach space ''B'' is called '''nuclear''' if it satisfies the condition above with all ''f_n^*'' bounded by 1 on some fixed neighborhood of 0.

{{reflist}}

==References==

* A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,''Mem. Am. Math.Soc.'' '''16'''. {{MR|0075539}}

* A. Grothendieck (1956), La theorie de Fredholm, ''Bull. Soc. Math. France'', '''84''':319–384. {{MR|0088665}}

* A. Hinrichs and A. Pietsch (2010), ''p''-nuclear operators in the sense of Grothendieck, ''Mathematische Nachrichen'' '''283''': 232–261. {{doi|10.1002/mana.200910128}}. {{MR|2604120}}

* {{springer |id=Nuclear_operator |author=G. L. Litvinov |title=Nuclear operator }}

* {{Citation |first1=H. H. |last1=Schaefer |first2=M. P. |last2=Wolff |title=Topological vector spaces |series=Graduate Texts in Mathematics |volume=3 |edition=2 |publisher=Springer |year=1999 |isbn=0-387-98726-6 |doi=10.1007/978-1-4612-1468-7}}

* {{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}}

[[Category:Operator theory]]

[[Category:Topological tensor products]]

[[Category:Linear operators]]