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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 notationEdit
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 topologiesEdit
- 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)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 bilinear 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 spacesEdit
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 . Explicitely,[clarification needed explicitly or especially?] if is a nuclear operator then .
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.
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 spacesEdit
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 and . 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 linear operators.
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 spacesEdit
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 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 .
- 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.
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 .
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.
- Auxiliary normed spaces
- Initial topology – coarsest topology making certain functions continuous
- Inductive tensor product
- 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 – Type of topological vector space
- Projective tensor product
- 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
- Topological vector space – Vector space with a notion of nearness
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