Nuclear operators between Banach spaces

In mathematics, nuclear operators between Banach spaces are a linear operators between Banach spaces in infinite dimensions that share some of the properties of their counter-part in finite dimension. In Hilbert spaces such operators are usually called trace class operators and one can define such things as the trace. In Banach spaces this is no longer possible for general nuclear operators, it is however possible for -nuclear operator via the Grothendieck trace theorem.

The general definition for Banach spaces was given by Grothendieck. This article presents both cases but concentrates on the general case of nuclear operators on Banach spaces.

Nuclear operators on Hilbert spaces

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An operator   on a Hilbert space     is compact if it can be written in the form[citation needed]   where   and   and   are (not necessarily complete) orthonormal sets. Here   is a set of real numbers, the set of singular values of the operator, obeying   if  

The bracket   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 nuclear or trace-class if  

Properties

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A nuclear operator on a Hilbert space has the important property that a trace operation may be defined. Given an orthonormal basis   for the Hilbert space, the trace is defined as  

Obviously, the sum converges absolutely, and it can be proven that the result is independent of the basis[citation needed]. It can be shown that this trace is identical to the sum of the eigenvalues of   (counted with multiplicity).

Nuclear operators on Banach spaces

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The definition of trace-class operator was extended to Banach spaces by Alexander Grothendieck in 1955.

Let   and   be Banach spaces, and   be the dual of   that is, the set of all continuous or (equivalently) bounded linear functionals on   with the usual norm. There is a canonical evaluation map   (from the projective tensor product of   and   to the Banach space of continuous linear maps from   to  ). It is determined by sending   and   to the linear map   An operator   is called nuclear if it is in the image of this evaluation map.[1]

q-nuclear operators

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An operator   is said to be nuclear of order   if there exist sequences of vectors   with   functionals   with   and complex numbers   with   such that the operator may be written as   with the sum converging in the operator norm.

Operators that are nuclear of order 1 are called nuclear operators: these are the ones for which the series   is absolutely convergent. Nuclear operators of order 2 are called Hilbert–Schmidt operators.

Relation to trace-class operators

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With additional steps, a trace may be defined for such operators when  

Properties

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The trace and determinant can no longer be defined in general in Banach spaces. However they can be defined for the so-called  -nuclear operators via Grothendieck trace theorem.

Generalizations

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More generally, an operator from a locally convex topological vector space   to a Banach space   is called nuclear if it satisfies the condition above with all   bounded by 1 on some fixed neighborhood of 0.

An extension of the concept of nuclear maps to arbitrary monoidal categories is given by Stolz & Teichner (2012). A monoidal category can be thought of as a 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   in a monoidal category is called thick if it can be written as a composition   for an appropriate object   and maps   where   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.[2]

Examples

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Suppose that   and   are Hilbert-Schmidt operators between Hilbert spaces. Then the composition   is a nuclear operator.[3]

See also

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References

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  1. ^ Schaefer & Wolff (1999, Chapter III, §7)
  2. ^ Stolz & Teichner (2012, Theorem 4.26)
  3. ^ Schaefer & Wolff 1999, p. 177.
  • A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,Mem. Am. Math.Soc. 16. MR0075539
  • A. Grothendieck (1956), La theorie de Fredholm, Bull. Soc. Math. France, 84:319–384. MR0088665
  • A. Hinrichs and A. Pietsch (2010), p-nuclear operators in the sense of Grothendieck, Mathematische Nachrichen 283: 232–261. doi:10.1002/mana.200910128. MR2604120
  • G. L. Litvinov (2001) [1994], "Nuclear operator", Encyclopedia of Mathematics, EMS Press
  • Schaefer, H. H.; Wolff, M. P. (1999), Topological vector spaces, Graduate Texts in Mathematics, vol. 3 (2 ed.), Springer, doi:10.1007/978-1-4612-1468-7, ISBN 0-387-98726-6
  • Stolz, Stephan; Teichner, Peter (2012), "Traces in monoidal categories", Transactions of the American Mathematical Society, 364 (8): 4425–4464, arXiv:1010.4527, doi:10.1090/S0002-9947-2012-05615-7, MR 2912459