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 notationEdit

Throughout let X,Y, and Z be topological vector spaces (TVSs) and L : XY 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 : XY 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 XX/S and the canonical injection SX are homomorphisms.
  • The set of continuous linear maps XZ (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 : HH from a Hilbert space into itself is called positive if   for every  . In this case, there is a unique positive map r : HH, 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 : HH 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 : HH 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]

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 XY then it can be shown that XY together with 𝜒 forms a tensor product of X and Y (where xy := 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(XY; Z) → Bi(X, Y; Z) given by uu𝜒 is an isomorphism of vector spaces. In particular, this allows us to identify the algebraic dual of XY with the space of bilinear forms on X × Y.[4] Moreover, if X and Y are locally convex topological vector spaces (TVSs) and if XY 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 XY 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]

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.[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  . Explicitely,[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:
  1.   is nuclear.
  2. 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]

Nuclear operators between Hilbert spacesEdit

Nuclear automorphisms of a Hilbert space are called trace class operators.

Let X and Y be Hilbert spaces and let N : XY be a continuous linear map. Suppose that   where R : XX is the square-root of   and U : XY 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.[10]

CharacterizationsEdit

Let X and Y be Hilbert spaces and let N : XY be a continuous linear map whose absolute value is R : XX. The following are equivalent:

  1. N : XY is nuclear.
  2. R : XX is nuclear.[11]
  3. R : XX 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  .
  4.   is nuclear, in which case  . [9]
  5. There are two orthogonal sequences   in X and   in Y, and a sequence   in   such that for all  ,  .[11]
  6. N : XY' is an integral map.[12]

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  .[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:
  1.   is nuclear.
  2. (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.[6]
  3. There exist Banach spaces   and   and continuous linear maps  ,  , and   such that   is nuclear and  .[8]
  4. 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]

PropertiesEdit

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  .[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.

  • Any nuclear map is compact.[2]
  • For every topology of uniform convergence on  , the nuclear maps are contained in the closure of   (when   is viewed as a subspace of  ).[6]

See alsoEdit

ReferencesEdit

  1. ^ Trèves 2006, p. 488.
  2. ^ a b c Trèves 2006, p. 483.
  3. ^ a b Trèves 2006, p. 490.
  4. ^ Schaefer & Wolff 1999, p. 92.
  5. ^ a b Schaefer & Wolff 1999, p. 93.
  6. ^ a b c Schaefer & Wolff 1999, p. 98.
  7. ^ a b c Trèves 2006, pp. 478-479.
  8. ^ a b c d e Trèves 2006, pp. 481-483.
  9. ^ a b c Trèves 2006, p. 484.
  10. ^ a b Trèves 2006, pp. 483-484.
  11. ^ a b c Trèves 2006, pp. 492-494.
  12. ^ Trèves 2006, pp. 502-508.
  13. ^ Trèves 2006, pp. 479-481.
  14. ^ a b Schaefer & Wolff 1999, p. 100.
  15. ^ a b Trèves 2006, p. 485.

BibliographyEdit

  • Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4440-3. OCLC 185095773.
  • Dubinsky, Ed (1979). The structure of nuclear Fréchet spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09504-7. OCLC 5126156.
  • Grothendieck, Alexander (1966). Produits tensoriels topologiques et espaces nucléaires (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. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (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 linksEdit