Nuclear operator
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).

Preliminaries and notation 

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).

 The projective tensor product of two locally convex TVSs X and Y is denoted by  and the completion of this space will be denoted by . 
 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 . 
 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.
 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: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.

Notation for topologies 

 σ(X, X′) denotes the coarsest topology on X making every map in X′ continuous and  or   denotes X endowed with this topology. 
 σ(X′, X) denotes weak-* topology on X* and  or   denotes X′ endowed with this topology. 
 Note that every  induces a map  defined by . σ(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  or  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  or  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) 

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. 
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. 
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 .

Nuclear operators between Banach spaces 

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. The range of this map is denoted by  and its elements are called nuclear operators.  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, if  is a nuclear operator then .

Characterization 
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:  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.
 If Y is reflexive then  is a nuclear if and only if  is nuclear, in which case .

Properties 
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 .

Nuclear operators between Hilbert spaces 

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  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.

Characterizations 

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.
R : X → X is compact and  is finite, in which case .
 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 .  
 is nuclear, in which case . 
There are two orthogonal sequences  in X and  in Y, and a sequence  in  such that for all , .
N : X → Y is an integral map.

Nuclear operators between locally convex 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  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 .

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.

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 
 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 . 
 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). 
 If in addition X and Y are Banach spaces, then .
 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.

Characterizations 
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 disk B 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.
 There exist Banach spaces  and  and continuous linear maps , , and  such that  is nuclear and .  
 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:  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 .

Properties 
The following is a type of Hahn-Banach theorem for extending nuclear maps:
 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 . 
 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 .

Let X and Y be Hausdorff locally convex spaces and let  be a continuous linear operator. 
 Any nuclear map is compact.
 For every topology of uniform convergence on , the nuclear maps are contained in the closure of  (when  is viewed as a subspace of ).

See also

References

Bibliography

External links 

 Nuclear space at ncatlab

Topological vector spaces
Tensors
Operator theory
Topological tensor products
Linear operators