Nuclear operators between Banach spaces: Difference between revisions

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== Compact 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 {{mvar|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> are a set of real numbers, the [[singular value]]s of the operator, obeying {{math|''ρ<submath>n\rho_n \to 0</submath>''}} → 0 if {{mvar|<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.

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

: <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 [[traceTrace class|trace]] operation may be defined. Given an orthonormal basis <math>\{\psi_n\}</math> for the Hilbert space, the trace is defined as

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

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).

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

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

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 {{mvar|<math>B}}</math> to the Banach space of continuous linear maps from {{mvar|<math>A}}</math> to {{mvar|<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 ===

An operator

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

is said to be '''{{em|nuclear of order {{mvar|<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.

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

With additional steps, a trace may be defined for such operators when {{<math|1=''>A'' = ''B''}}.</math>

=== Generalizations ===

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

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

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

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

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

A monoidal category can be thought of as a [[categoryCategory (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 {{mvar|<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_2</math> is a [[nuclear operator]].{{sfn|Schaefer|Wolff|1999|p=177}}

== References ==

{{Reflistreflist}}

<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}}