Throughout let X,Y, and Z be topological vector spaces (TVSs) and L : X → Y be a linear operator (no assumption of continuity is made unless otherwise stated).
L : X → Y is a topological homomorphism or homomorphism, if it is linear, continuous, and is an open map, where , the image of L, has the subspace topology induced by Y.
If S is a subspace of X then both the quotient map X → X/S and the canonical injection S → X are homomorphisms.
The set of continuous linear maps X → Z (resp. continuous bilinear maps ) will be denoted by L(X, Z) (resp. B(X, Y; Z)) where if Z is the underlying scalar field then we may instead write L(X) (resp. B(X, Y)).
Any linear map can be canonically decomposed as follows: where defines a bijection called the canonical bijection associated with L.
X* or will denote the continuous dual space of X.
To increase the clarity of the exposition, we use the common convention of writing elements of with a prime following the symbol (e.g. denotes an element of and not, say, a derivative and the variables x and need not be related in any way).
will denote the algebraic dual space of X (which is the vector space of all linear functionals on X, whether continuous or not).
A linear map L : H → H from a Hilbert space into itself is called positive if for every . In this case, there is a unique positive map r : H → H, called the square-root of L, such that .[1]
If is any continuous linear map between Hilbert spaces, then is always positive. Now let R : H → H denote its positive square-root, which is called the absolute value of L. Define first on by setting for and extending continuously to , and then define U on by setting for and extend this map linearly to all of . The map is a surjective isometry and .
A linear map is called compact or completely continuous if there is a neighborhood U of the origin in X such that is precompact in Y.[2]
In a Hilbert space, positive compact linear operators, say L : H → H have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:[3]
There is a sequence of positive numbers, decreasing and either finite or else converging to 0, and a sequence of nonzero finite dimensional subspaces of H (i = 1, 2, ) with the following properties: (1) the subspaces are pairwise orthogonal; (2) for every i and every , ; and (3) the orthogonal of the subspace spanned by is equal to the kernel of L.[3]
As usual, if X* is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be b(X′, X).
A canonical tensor product as a subspace of the dual of Bi(X, Y)Edit
Let X and Y be vector spaces (no topology is needed yet) and let Bi(X, Y) be the space of all bilinear maps defined on and going into the underlying scalar field.
For every , let be the canonical linear form on Bi(X, Y) defined by for every u ∈ Bi(X, Y). This induces a canonical map defined by , where denotes the algebraic dual of Bi(X, Y). If we denote the span of the range of 𝜒 by X ⊗ Y then it can be shown that X ⊗ Y together with 𝜒 forms a tensor product of X and Y (where x ⊗ y := 𝜒(x, y)). This gives us a canonical tensor product of X and Y.
If Z is any other vector space then the mapping Li(X ⊗ Y; Z) → Bi(X, Y; Z) given by u ↦ u ∘ 𝜒 is an isomorphism of vector spaces. In particular, this allows us to identify the algebraic dual of X ⊗ Y with the space of bilinear forms on X × Y.[4] Moreover, if X and Y are locally convex topological vector spaces (TVSs) and if X ⊗ Y is given the 𝜋-topology then for every locally convex TVS Z, this map restricts to a vector space isomorphism from the space of continuous linear mappings onto the space of continuous bilinear mappings.[5] In particular, the continuous dual of X ⊗ Y can be canonically identified with the space B(X, Y) of continuous bilinear forms on X × Y; furthermore, under this identification the equicontinuous subsets of B(X, Y) are the same as the equicontinuous subsets of .[5]
There is a canonical vector space embedding defined by sending to the map
Assuming that X and Y are Banach spaces, then the map has norm (to see that the norm is , note that so that ). Thus it has a continuous extension to a map , where it is known that this map is not necessarily injective.[6] The range of this map is denoted by and its elements are called nuclear operators.[7] is TVS-isomorphic to and the norm on this quotient space, when transferred to elements of via the induced map , is called the trace-norm and is denoted by . Explicitly,[clarification needed explicitly or especially?] if is a nuclear operator then .
CharacterizationEdit
Suppose that X and Y are Banach spaces and that is a continuous linear operator.
The following are equivalent:
is nuclear.
There exists an sequence in the closed unit ball of , a sequence in the closed unit ball of , and a complex sequence such that and is equal to the mapping:[8] for all . Furthermore, the trace-norm is equal to the infimum of the numbers over the set of all representations of as such a series.[8]
If Y is reflexive then is a nuclear if and only if is nuclear, in which case . [9]
PropertiesEdit
Let X and Y be Banach spaces and let be a continuous linear operator.
If is a nuclear map then its transpose is a continuous nuclear map (when the dual spaces carry their strong dual topologies) and .[10]
Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map. Suppose that where R : X → X is the square-root of and U : X → Y is such that is a surjective isometry. Then N is a nuclear map if and only if R is a nuclear map; hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive self-adjoint operators.[10]
CharacterizationsEdit
Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map whose absolute value is R : X → X. The following are equivalent:
R : X → X is compact and is finite, in which case .[11]
Here, is the trace of R and it is defined as follows: Since R is a continuous compact positive operator, there exists a (possibly finite) sequence of positive numbers with corresponding non-trivial finite-dimensional and mutually orthogonal vector spaces such that the orthogonal (in H) of is equal to (and hence also to ) and for all k, for all ; the trace is defined as .
Suppose that U is a convex balanced closed neighborhood of the origin in X and B is a convex balanced bounded Banach disk in Y with both X and Y locally convex spaces. Let and let be the canonical projection. One can define the auxiliary Banach space with the canonical map whose image, , is dense in as well as the auxiliary space normed by and with a canonical map being the (continuous) canonical injection. Given any continuous linear map one obtains through composition the continuous linear map ; thus we have an injection and we henceforth use this map to identify as a subspace of .[7]
Definition: Let X and Y be Hausdorff locally convex spaces. The union of all as U ranges over all closed convex balanced neighborhoods of the origin in X and B ranges over all bounded Banach disks in Y, is denoted by and its elements are call nuclear mappings of X into Y.[7]
When X and Y are Banach spaces, then this new definition of nuclear mapping is consistent with the original one given for the special case where X and Y are Banach spaces.
Sufficient conditions for nuclearityEdit
Let W, X, Y, and Z be Hausdorff locally convex spaces, a nuclear map, and and be continuous linear maps. Then , , and are nuclear and if in addition W, X, Y, and Z are all Banach spaces then .[13][14]
If is a nuclear map between two Hausdorff locally convex spaces, then its transpose is a continuous nuclear map (when the dual spaces carry their strong dual topologies).[2]
If in addition X and Y are Banach spaces, then .[9]
If is a nuclear map between two Hausdorff locally convex spaces and if is a completion of X, then the unique continuous extension of N is nuclear.[14]
CharacterizationsEdit
Let X and Y be Hausdorff locally convex spaces and let be a continuous linear operator.
The following are equivalent:
is nuclear.
(Definition) There exists a convex balanced neighborhood U of the origin in X and a bounded Banach diskB in Y such that and the induced map is nuclear, where is the unique continuous extension of , which is the unique map satisfying where is the natural inclusion and is the canonical projection.[6]
There exist Banach spaces and and continuous linear maps , , and such that is nuclear and .[8]
There exists an equicontinuous sequence in , a bounded Banach disk, a sequence in B, and a complex sequence such that and is equal to the mapping:[8] for all .
If X is barreled and Y is quasi-complete, then N is nuclear if and only if N has a representation of the form with bounded in , bounded in Y and .[8]
If is a TVS-embedding and is a nuclear map then there exists a nuclear map such that . Furthermore, when X and Y are Banach spaces and E is an isometry then for any , can be picked so that .[15]
Suppose that is a TVS-embedding whose image is closed in Z and let be the canonical projection. Suppose all that every compact disk in is the image under of a bounded Banach disk in Z (this is true, for instance, if X and Z are both Fréchet spaces, or if Z is the strong dual of a Fréchet space and is weakly closed in Z). Then for every nuclear map there exists a nuclear map such that .
Furthermore, when X and Z are Banach spaces and E is an isometry then for any , can be picked so that .[15]
Let X and Y be Hausdorff locally convex spaces and let be a continuous linear operator.
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External linksEdit
Nuclear space at ncatlab
September 01, 2023
nuclear, operator, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, june, 20. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Nuclear operator news newspapers books scholar JSTOR June 2020 Learn how and when to remove this template message In mathematics nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces TVSs Contents 1 Preliminaries and notation 1 1 Notation for topologies 1 2 A canonical tensor product as a subspace of the dual of Bi X Y 2 Nuclear operators between Banach spaces 2 1 Characterization 2 2 Properties 3 Nuclear operators between Hilbert spaces 3 1 Characterizations 4 Nuclear operators between locally convex spaces 4 1 Sufficient conditions for nuclearity 4 2 Characterizations 4 3 Properties 5 See also 6 References 7 Bibliography 8 External linksPreliminaries and notation EditThroughout let X Y and Z be topological vector spaces TVSs and L X Y be a linear operator no assumption of continuity is made unless otherwise stated The projective tensor product of two locally convex TVSs X and Y is denoted by X p Y displaystyle X otimes pi Y nbsp and the completion of this space will be denoted by X p Y displaystyle X widehat otimes pi Y nbsp L X Y is a topological homomorphism or homomorphism if it is linear continuous and L X Im L displaystyle L X to operatorname Im L nbsp is an open map where Im L displaystyle operatorname Im L nbsp the image of L has the subspace topology induced by Y If S is a subspace of X then both the quotient map X X S and the canonical injection S X are homomorphisms The set of continuous linear maps X Z resp continuous bilinear maps X Y Z displaystyle X times Y to Z nbsp will be denoted by L X Z resp B X Y Z where if Z is the underlying scalar field then we may instead write L X resp B X Y Any linear map L X Y displaystyle L X to Y nbsp can be canonically decomposed as follows X X ker L L 0 Im L Y displaystyle X to X ker L xrightarrow L 0 operatorname Im L to Y nbsp where L 0 x ker L L x displaystyle L 0 left x ker L right L x nbsp defines a bijection called the canonical bijection associated with L X or X displaystyle X nbsp will denote the continuous dual space of X To increase the clarity of the exposition we use the common convention of writing elements of X displaystyle X nbsp with a prime following the symbol e g x displaystyle x nbsp denotes an element of X displaystyle X nbsp and not say a derivative and the variables x and x displaystyle x nbsp need not be related in any way X displaystyle X nbsp will denote the algebraic dual space of X which is the vector space of all linear functionals on X whether continuous or not A linear map L H H from a Hilbert space into itself is called positive if L x x 0 displaystyle langle L x x rangle geq 0 nbsp for every x H displaystyle x in H nbsp In this case there is a unique positive map r H H called the square root of L such that L r r displaystyle L r circ r nbsp 1 If L H 1 H 2 displaystyle L H 1 to H 2 nbsp is any continuous linear map between Hilbert spaces then L L displaystyle L circ L nbsp is always positive Now let R H H denote its positive square root which is called the absolute value of L Define U H 1 H 2 displaystyle U H 1 to H 2 nbsp first on Im R displaystyle operatorname Im R nbsp by setting U x L x displaystyle U x L x nbsp for x R x 1 Im R displaystyle x R left x 1 right in operatorname Im R nbsp and extending U displaystyle U nbsp continuously to Im R displaystyle overline operatorname Im R nbsp and then define U on ker R displaystyle ker R nbsp by setting U x 0 displaystyle U x 0 nbsp for x ker R displaystyle x in ker R nbsp and extend this map linearly to all of H 1 displaystyle H 1 nbsp The map U Im R Im R Im L displaystyle U big vert operatorname Im R operatorname Im R to operatorname Im L nbsp is a surjective isometry and L U R displaystyle L U circ R nbsp A linear map L X Y displaystyle Lambda X to Y nbsp is called compact or completely continuous if there is a neighborhood U of the origin in X such that L U displaystyle Lambda U nbsp is precompact in Y 2 In a Hilbert space positive compact linear operators say L H H have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F Riesz 3 There is a sequence of positive numbers decreasing and either finite or else converging to 0 r 1 gt r 2 gt gt r k gt displaystyle r 1 gt r 2 gt cdots gt r k gt cdots nbsp and a sequence of nonzero finite dimensional subspaces V i displaystyle V i nbsp of H i 1 2 displaystyle ldots nbsp with the following properties 1 the subspaces V i displaystyle V i nbsp are pairwise orthogonal 2 for every i and every x V i displaystyle x in V i nbsp L x r i x displaystyle L x r i x nbsp and 3 the orthogonal of the subspace spanned by i V i textstyle bigcup i V i nbsp is equal to the kernel of L 3 Notation for topologies Edit Main articles Topology of uniform convergence and Mackey topology s X X denotes the coarsest topology on X making every map in X continuous and X s X X displaystyle X sigma left X X right nbsp or X s displaystyle X sigma nbsp denotes X endowed with this topology s X X denotes weak topology on X and X s X X displaystyle X sigma left X X right nbsp or X s displaystyle X sigma nbsp denotes X endowed with this topology Note that every x 0 X displaystyle x 0 in X nbsp induces a map X R displaystyle X to mathbb R nbsp defined by l l x 0 displaystyle lambda mapsto lambda left x 0 right nbsp s X X is the coarsest topology on X making all such maps continuous b X X denotes the topology of bounded convergence on X and X b X X displaystyle X b left X X right nbsp or X b displaystyle X b nbsp denotes X endowed with this topology b X X denotes the topology of bounded convergence on X or the strong dual topology on X and X b X X displaystyle X b left X X right nbsp or X b displaystyle X b nbsp denotes X endowed with this topology As usual if X is considered as a topological vector space but it has not been made clear what topology it is endowed with then the topology will be assumed to be b X X A canonical tensor product as a subspace of the dual of Bi X Y Edit Let X and Y be vector spaces no topology is needed yet and let Bi X Y be the space of all bilinear maps defined on X Y displaystyle X times Y nbsp and going into the underlying scalar field For every x y X Y displaystyle x y in X times Y nbsp let x x y displaystyle chi x y nbsp be the canonical linear form on Bi X Y defined by x x y u u x y displaystyle chi x y u u x y nbsp for every u Bi X Y This induces a canonical map x X Y B i X Y displaystyle chi X times Y to mathrm Bi X Y nbsp defined by x x y x x y displaystyle chi x y chi x y nbsp where B i X Y displaystyle mathrm Bi X Y nbsp denotes the algebraic dual of Bi X Y If we denote the span of the range of 𝜒 by X Y then it can be shown that X Y together with 𝜒 forms a tensor product of X and Y where x y 𝜒 x y This gives us a canonical tensor product of X and Y If Z is any other vector space then the mapping Li X Y Z Bi X Y Z given by u u 𝜒 is an isomorphism of vector spaces In particular this allows us to identify the algebraic dual of X Y with the space of bilinear forms on X Y 4 Moreover if X and Y are locally convex topological vector spaces TVSs and if X Y is given the 𝜋 topology then for every locally convex TVS Z this map restricts to a vector space isomorphism L X p Y Z B X Y Z displaystyle L X otimes pi Y Z to B X Y Z nbsp from the space of continuous linear mappings onto the space of continuous bilinear mappings 5 In particular the continuous dual of X Y can be canonically identified with the space B X Y of continuous bilinear forms on X Y furthermore under this identification the equicontinuous subsets of B X Y are the same as the equicontinuous subsets of X p Y displaystyle X otimes pi Y nbsp 5 Nuclear operators between Banach spaces EditMain articles Nuclear operators between Banach spaces and Projective tensor product There is a canonical vector space embedding I X Y L X Y displaystyle I X otimes Y to L X Y nbsp defined by sending z i n x i y i textstyle z sum i n x i otimes y i nbsp to the map x i n x i x y i displaystyle x mapsto sum i n x i x y i nbsp Assuming that X and Y are Banach spaces then the map I X b p Y L b X Y displaystyle I X b otimes pi Y to L b X Y nbsp has norm 1 displaystyle 1 nbsp to see that the norm is 1 displaystyle leq 1 nbsp note that I z sup x 1 I z x sup x 1 i 1 n x i x y i sup x 1 i 1 n x i x y i i 1 n x i y i textstyle I z sup x leq 1 I z x sup x leq 1 left sum i 1 n x i x y i right leq sup x leq 1 sum i 1 n left x i right x left y i right leq sum i 1 n left x i right left y i right nbsp so that I z z p displaystyle left I z right leq left z right pi nbsp Thus it has a continuous extension to a map I X b p Y L b X Y displaystyle hat I X b widehat otimes pi Y to L b X Y nbsp where it is known that this map is not necessarily injective 6 The range of this map is denoted by L 1 X Y displaystyle L 1 X Y nbsp and its elements are called nuclear operators 7 L 1 X Y displaystyle L 1 X Y nbsp is TVS isomorphic to X b p Y ker I displaystyle left X b widehat otimes pi Y right ker hat I nbsp and the norm on this quotient space when transferred to elements of L 1 X Y displaystyle L 1 X Y nbsp via the induced map I X b p Y ker I L 1 X Y displaystyle hat I left X b widehat otimes pi Y right ker hat I to L 1 X Y nbsp is called the trace norm and is denoted by Tr displaystyle cdot operatorname Tr nbsp Explicitly clarification needed explicitly or especially if T X Y displaystyle T X to Y nbsp is a nuclear operator then T Tr inf z I 1 T z p textstyle left T right operatorname Tr inf z in hat I 1 left T right left z right pi nbsp Characterization Edit Suppose that X and Y are Banach spaces and that N X Y displaystyle N X to Y nbsp is a continuous linear operator The following are equivalent N X Y displaystyle N X to Y nbsp is nuclear There exists an sequence x i i 1 displaystyle left x i right i 1 infty nbsp in the closed unit ball of X displaystyle X nbsp a sequence y i i 1 displaystyle left y i right i 1 infty nbsp in the closed unit ball of Y displaystyle Y nbsp and a complex sequence c i i 1 displaystyle left c i right i 1 infty nbsp such that i 1 c i lt textstyle sum i 1 infty c i lt infty nbsp and N displaystyle N nbsp is equal to the mapping 8 N x i 1 c i x i x y i textstyle N x sum i 1 infty c i x i x y i nbsp for all x X displaystyle x in X nbsp Furthermore the trace norm N Tr displaystyle N operatorname Tr nbsp is equal to the infimum of the numbers i 1 c i textstyle sum i 1 infty c i nbsp over the set of all representations of N displaystyle N nbsp as such a series 8 If Y is reflexive then N X Y displaystyle N X to Y nbsp is a nuclear if and only if t N Y b X b displaystyle t N Y b to X b nbsp is nuclear in which case t N Tr N Tr textstyle left t N right operatorname Tr left N right operatorname Tr nbsp 9 Properties Edit Let X and Y be Banach spaces and let N X Y displaystyle N X to Y nbsp be a continuous linear operator If N X Y displaystyle N X to Y nbsp is a nuclear map then its transpose t N Y b X b displaystyle t N Y b to X b nbsp is a continuous nuclear map when the dual spaces carry their strong dual topologies and t N Tr N Tr textstyle left t N right operatorname Tr leq left N right operatorname Tr nbsp 10 Nuclear operators between Hilbert spaces EditSee also Trace class Nuclear automorphisms of a Hilbert space are called trace class operators Let X and Y be Hilbert spaces and let N X Y be a continuous linear map Suppose that N U R displaystyle N UR nbsp where R X X is the square root of N N displaystyle N N nbsp and U X Y is such that U Im R Im R Im N displaystyle U big vert operatorname Im R operatorname Im R to operatorname Im N nbsp is a surjective isometry Then N is a nuclear map if and only if R is a nuclear map hence to study nuclear maps between Hilbert spaces it suffices to restrict one s attention to positive self adjoint operators 10 Characterizations Edit Let X and Y be Hilbert spaces and let N X Y be a continuous linear map whose absolute value is R X X The following are equivalent N X Y is nuclear R X X is nuclear 11 R X X is compact and Tr R displaystyle operatorname Tr R nbsp is finite in which case Tr R N Tr displaystyle operatorname Tr R N operatorname Tr nbsp 11 Here Tr R displaystyle operatorname Tr R nbsp is the trace of R and it is defined as follows Since R is a continuous compact positive operator there exists a possibly finite sequence l 1 gt l 2 gt displaystyle lambda 1 gt lambda 2 gt cdots nbsp of positive numbers with corresponding non trivial finite dimensional and mutually orthogonal vector spaces V 1 V 2 displaystyle V 1 V 2 ldots nbsp such that the orthogonal in H of span V 1 V 2 displaystyle operatorname span left V 1 cup V 2 cup cdots right nbsp is equal to ker R displaystyle ker R nbsp and hence also to ker N displaystyle ker N nbsp and for all k R x l k x displaystyle R x lambda k x nbsp for all x V k displaystyle x in V k nbsp the trace is defined as Tr R k l k dim V k textstyle operatorname Tr R sum k lambda k dim V k nbsp t N Y b X b displaystyle t N Y b to X b nbsp is nuclear in which case t N Tr N Tr displaystyle t N operatorname Tr N operatorname Tr nbsp 9 There are two orthogonal sequences x i i 1 displaystyle left x i right i 1 infty nbsp in X and y i i 1 displaystyle left y i right i 1 infty nbsp in Y and a sequence l i i 1 displaystyle left lambda i right i 1 infty nbsp in l 1 displaystyle l 1 nbsp such that for all x X displaystyle x in X nbsp N x i l i x x i y i textstyle N x sum i lambda i langle x x i rangle y i nbsp 11 N X Y is an integral map 12 Nuclear operators between locally convex spaces EditSee also Auxiliary normed spaces Suppose that U is a convex balanced closed neighborhood of the origin in X and B is a convex balanced bounded Banach disk in Y with both X and Y locally convex spaces Let p U x inf r gt 0 x r U r textstyle p U x inf r gt 0 x in rU r nbsp and let p X X p U 1 0 displaystyle pi X to X p U 1 0 nbsp be the canonical projection One can define the auxiliary Banach space X U displaystyle hat X U nbsp with the canonical map p U X X U displaystyle hat pi U X to hat X U nbsp whose image X p U 1 0 displaystyle X p U 1 0 nbsp is dense in X U displaystyle hat X U nbsp as well as the auxiliary space F B span B displaystyle F B operatorname span B nbsp normed by p B y inf r gt 0 y r B r textstyle p B y inf r gt 0 y in rB r nbsp and with a canonical map i F B F displaystyle iota F B to F nbsp being the continuous canonical injection Given any continuous linear map T X U Y B displaystyle T hat X U to Y B nbsp one obtains through composition the continuous linear map p U T i X Y displaystyle hat pi U circ T circ iota X to Y nbsp thus we have an injection L X U Y B L X Y textstyle L left hat X U Y B right to L X Y nbsp and we henceforth use this map to identify L X U Y B textstyle L left hat X U Y B right nbsp as a subspace of L X Y displaystyle L X Y nbsp 7 Definition Let X and Y be Hausdorff locally convex spaces The union of all L 1 X U Y B textstyle L 1 left hat X U Y B right nbsp as U ranges over all closed convex balanced neighborhoods of the origin in X and B ranges over all bounded Banach disks in Y is denoted by L 1 X Y displaystyle L 1 X Y nbsp and its elements are call nuclear mappings of X into Y 7 When X and Y are Banach spaces then this new definition of nuclear mapping is consistent with the original one given for the special case where X and Y are Banach spaces Sufficient conditions for nuclearity Edit Let W X Y and Z be Hausdorff locally convex spaces N X Y displaystyle N X to Y nbsp a nuclear map and M W X displaystyle M W to X nbsp and P Y Z displaystyle P Y to Z nbsp be continuous linear maps Then N M W Y displaystyle N circ M W to Y nbsp P N X Z displaystyle P circ N X to Z nbsp and P N M W Z displaystyle P circ N circ M W to Z nbsp are nuclear and if in addition W X Y and Z are all Banach spaces then P N M Tr P N Tr M textstyle left P circ N circ M right operatorname Tr leq left P right left N right operatorname Tr left M right nbsp 13 14 If N X Y displaystyle N X to Y nbsp is a nuclear map between two Hausdorff locally convex spaces then its transpose t N Y b X b displaystyle t N Y b to X b nbsp is a continuous nuclear map when the dual spaces carry their strong dual topologies 2 If in addition X and Y are Banach spaces then t N Tr N Tr textstyle left t N right operatorname Tr leq left N right operatorname Tr nbsp 9 If N X Y displaystyle N X to Y nbsp is a nuclear map between two Hausdorff locally convex spaces and if X displaystyle hat X nbsp is a completion of X then the unique continuous extension N X Y displaystyle hat N hat X to Y nbsp of N is nuclear 14 Characterizations Edit Let X and Y be Hausdorff locally convex spaces and let N X Y displaystyle N X to Y nbsp be a continuous linear operator The following are equivalent N X Y displaystyle N X to Y nbsp is nuclear Definition There exists a convex balanced neighborhood U of the origin in X and a bounded Banach disk B in Y such that N U B displaystyle N U subseteq B nbsp and the induced map N 0 X U Y B displaystyle overline N 0 hat X U to Y B nbsp is nuclear where N 0 displaystyle overline N 0 nbsp is the unique continuous extension of N 0 X U Y B displaystyle N 0 X U to Y B nbsp which is the unique map satisfying N In B N 0 p U displaystyle N operatorname In B circ N 0 circ pi U nbsp where In B Y B Y displaystyle operatorname In B Y B to Y nbsp is the natural inclusion and p U X X p U 1 0 displaystyle pi U X to X p U 1 0 nbsp is the canonical projection 6 There exist Banach spaces B 1 displaystyle B 1 nbsp and B 2 displaystyle B 2 nbsp and continuous linear maps f X B 1 displaystyle f X to B 1 nbsp n B 1 B 2 displaystyle n B 1 to B 2 nbsp and g B 2 Y displaystyle g B 2 to Y nbsp such that n B 1 B 2 displaystyle n B 1 to B 2 nbsp is nuclear and N g n f displaystyle N g circ n circ f nbsp 8 There exists an equicontinuous sequence x i i 1 displaystyle left x i right i 1 infty nbsp in X displaystyle X nbsp a bounded Banach disk B Y displaystyle B subseteq Y nbsp a sequence y i i 1 displaystyle left y i right i 1 infty nbsp in B and a complex sequence c i i 1 displaystyle left c i right i 1 infty nbsp such that i 1 c i lt textstyle sum i 1 infty c i lt infty nbsp and N displaystyle N nbsp is equal to the mapping 8 N x i 1 c i x i x y i textstyle N x sum i 1 infty c i x i x y i nbsp for all x X displaystyle x in X nbsp If X is barreled and Y is quasi complete then N is nuclear if and only if N has a representation of the form N x i 1 c i x i x y i textstyle N x sum i 1 infty c i x i x y i nbsp with x i i 1 displaystyle left x i right i 1 infty nbsp bounded in X displaystyle X nbsp y i i 1 displaystyle left y i right i 1 infty nbsp bounded in Y and i 1 c i lt textstyle sum i 1 infty c i lt infty nbsp 8 Properties Edit The following is a type of Hahn Banach theorem for extending nuclear maps If E X Z displaystyle E X to Z nbsp is a TVS embedding and N X Y displaystyle N X to Y nbsp is a nuclear map then there exists a nuclear map N Z Y displaystyle tilde N Z to Y nbsp such that N E N displaystyle tilde N circ E N nbsp Furthermore when X and Y are Banach spaces and E is an isometry then for any ϵ gt 0 displaystyle epsilon gt 0 nbsp N displaystyle tilde N nbsp can be picked so that N Tr N Tr ϵ displaystyle tilde N operatorname Tr leq N operatorname Tr epsilon nbsp 15 Suppose that E X Z displaystyle E X to Z nbsp is a TVS embedding whose image is closed in Z and let p Z Z Im E displaystyle pi Z to Z operatorname Im E nbsp be the canonical projection Suppose all that every compact disk in Z Im E displaystyle Z operatorname Im E nbsp is the image under p displaystyle pi nbsp of a bounded Banach disk in Z this is true for instance if X and Z are both Frechet spaces or if Z is the strong dual of a Frechet space and Im E displaystyle operatorname Im E nbsp is weakly closed in Z Then for every nuclear map N Y Z Im E displaystyle N Y to Z operatorname Im E nbsp there exists a nuclear map N Y Z displaystyle tilde N Y to Z nbsp such that p N N displaystyle pi circ tilde N N nbsp Furthermore when X and Z are Banach spaces and E is an isometry then for any ϵ gt 0 displaystyle epsilon gt 0 nbsp N displaystyle tilde N nbsp can be picked so that N Tr N Tr ϵ textstyle left tilde N right operatorname Tr leq left N right operatorname Tr epsilon nbsp 15 Let X and Y be Hausdorff locally convex spaces and let N X Y displaystyle N X to Y nbsp be a continuous linear operator Any nuclear map is compact 2 For every topology of uniform convergence on L X Y displaystyle L X Y nbsp the nuclear maps are contained in the closure of X Y displaystyle X otimes Y nbsp when X Y displaystyle X otimes Y nbsp is viewed as a subspace of L X Y displaystyle L X Y nbsp 6 See also EditAuxiliary normed spaces Covariance operator Operator in probability theory Initial topology Coarsest topology making certain functions continuous Inductive tensor product binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback Injective tensor product Locally convex topological vector space A vector space with a topology defined by convex open sets Nuclear operators between Banach spaces Nuclear space A generalization of finite dimensional Euclidean spaces different from Hilbert spaces Projective tensor product tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback Tensor product of Hilbert spaces Tensor product space endowed with a special inner product Topological tensor product Tensor product constructions for topological vector spaces Trace class Compact operator for which a finite trace can be defined Topological vector space Vector space with a notion of nearnessReferences Edit Treves 2006 p 488 a b c Treves 2006 p 483 a b Treves 2006 p 490 Schaefer amp Wolff 1999 p 92 a b Schaefer amp Wolff 1999 p 93 a b c Schaefer amp Wolff 1999 p 98 a b c Treves 2006 pp 478 479 a b c d e Treves 2006 pp 481 483 a b c Treves 2006 p 484 a b Treves 2006 pp 483 484 a b c Treves 2006 pp 492 494 Treves 2006 pp 502 508 Treves 2006 pp 479 481 a b Schaefer amp Wolff 1999 p 100 a b Treves 2006 p 485 Bibliography EditDiestel Joe 2008 The metric theory of tensor products Grothendieck s resume revisited Providence R I American Mathematical Society ISBN 978 0 8218 4440 3 OCLC 185095773 Dubinsky Ed 1979 The structure of nuclear Frechet spaces Berlin New York Springer Verlag ISBN 3 540 09504 7 OCLC 5126156 Grothendieck Alexander 1966 Produits tensoriels topologiques et espaces nucleaires in French Providence American Mathematical Society ISBN 0 8218 1216 5 OCLC 1315788 Husain Taqdir 1978 Barrelledness in topological and ordered vector spaces Berlin New York Springer Verlag ISBN 3 540 09096 7 OCLC 4493665 Khaleelulla S M 1982 Counterexamples in Topological Vector Spaces Lecture Notes in Mathematics Vol 936 Berlin Heidelberg New York Springer Verlag ISBN 978 3 540 11565 6 OCLC 8588370 Narici Lawrence Beckenstein Edward 2011 Topological Vector Spaces Pure and applied mathematics Second ed Boca Raton FL CRC Press ISBN 978 1584888666 OCLC 144216834 Nlend H 1977 Bornologies and functional analysis introductory course on the theory of duality topology bornology and its use in functional analysis Amsterdam New York New York North Holland Pub Co Sole distributors for the U S A and Canada Elsevier North Holland ISBN 0 7204 0712 5 OCLC 2798822 Nlend H 1981 Nuclear and conuclear spaces introductory courses on nuclear and conuclear spaces in the light of the duality Amsterdam New York New York N Y North Holland Pub Co Sole distributors for the U S A and Canada Elsevier North Holland ISBN 0 444 86207 2 OCLC 7553061 Pietsch Albrecht 1972 Nuclear locally convex spaces Berlin New York Springer Verlag ISBN 0 387 05644 0 OCLC 539541 Robertson A P 1973 Topological vector spaces Cambridge England University Press ISBN 0 521 29882 2 OCLC 589250 Ryan Raymond 2002 Introduction to tensor products of Banach spaces London New York Springer ISBN 1 85233 437 1 OCLC 48092184 Schaefer Helmut H Wolff Manfred P 1999 Topological Vector Spaces GTM Vol 8 Second ed New York NY Springer New York Imprint Springer ISBN 978 1 4612 7155 0 OCLC 840278135 Treves Francois 2006 1967 Topological Vector Spaces Distributions and Kernels Mineola N Y Dover Publications ISBN 978 0 486 45352 1 OCLC 853623322 Wong 1979 Schwartz spaces nuclear spaces and tensor products Berlin New York Springer Verlag ISBN 3 540 09513 6 OCLC 5126158 External links EditNuclear space at ncatlab Retrieved from https en wikipedia org w index php title Nuclear operator amp oldid 1142203467, wikipedia, 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