Nuclear operators between Banach spaces

In mathematics, a nuclear operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace). Nuclear operators are essentially the same as trace-class operators, though most authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces.

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; for more details about the important special case of nuclear (= trace-class) operators on Hilbert space, see the article Trace class.

Compact operator

An operator ${\displaystyle {\mathcal {L}}}$  on a Hilbert space ${\displaystyle {\mathcal {H}}}$  $${\displaystyle {\mathcal {L}}:{\mathcal {H}}\to {\mathcal {H}}}$$  is compact if it can be written in the form^{[citation needed]} $${\displaystyle {\mathcal {L}}=\sum _{n=1}^{N}\rho _{n}\langle f_{n},\cdot \rangle g_{n},}$$  where ${\displaystyle 1\leq N\leq \infty ,}$  and ${\displaystyle f_{1},\ldots ,f_{N}}$  and ${\displaystyle g_{1},\ldots ,g_{N}}$  are (not necessarily complete) orthonormal sets. Here ${\displaystyle \rho _{1},\ldots ,\rho _{N}}$  are a set of real numbers, the singular values of the operator, obeying ${\displaystyle \rho _{n}\to 0}$  if ${\displaystyle N=\infty .}$ 

The bracket ${\displaystyle \langle \cdot ,\cdot \rangle }$  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 $${\displaystyle \sum _{n=1}^{\infty }|\rho _{n}|<\infty .}$$ 

Properties

A nuclear operator on a Hilbert space has the important property that a trace operation may be defined. Given an orthonormal basis ${\displaystyle \{\psi _{n}\}}$  for the Hilbert space, the trace is defined as $${\displaystyle \operatorname {Tr} {\mathcal {L}}=\sum _{n}\langle \psi _{n},{\mathcal {L}}\psi _{n}\rangle .}$$ 

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 ${\displaystyle {\mathcal {L}}}$  (counted with multiplicity).

On Banach spaces

Main article: Fredholm kernel

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

Let ${\displaystyle A}$  and ${\displaystyle B}$  be Banach spaces, and ${\displaystyle A}$  be the dual of ${\displaystyle A,}$  that is, the set of all continuous or (equivalently) bounded linear functionals on ${\displaystyle A}$  with the usual norm. There is a canonical evaluation map $${\displaystyle A^{\prime }\otimes B\to \operatorname {Hom} (A,B)}$$  (from the projective tensor product of ${\displaystyle A}$  and ${\displaystyle B}$  to the Banach space of continuous linear maps from ${\displaystyle A}$  to ${\displaystyle B}$ ). It is determined by sending ${\displaystyle f\in A^{\prime }}$  and ${\displaystyle b\in B}$  to the linear map ${\displaystyle a\mapsto f(a)\cdot b.}$  An operator ${\displaystyle {\mathcal {L}}\in \operatorname {Hom} (A,B)}$  is called nuclear if it is in the image of this evaluation map.^[1]

q-nuclear operators

An operator $${\displaystyle {\mathcal {L}}:A\to B}$$  is said to be nuclear of order ${\displaystyle q}$  if there exist sequences of vectors ${\displaystyle \{g_{n}\}\in B}$  with ${\displaystyle \Vert g_{n}\Vert \leq 1,}$  functionals ${\displaystyle \left\{f_{n}^{*}\right\}\in A^{\prime }}$  with ${\displaystyle \Vert f_{n}^{*}\Vert \leq 1}$  and complex numbers ${\displaystyle \{\rho _{n}\}}$  with $${\displaystyle \sum _{n}|\rho _{n}|^{q}<\infty ,}$$  such that the operator may be written as $${\displaystyle {\mathcal {L}}=\sum _{n}\rho _{n}f_{n}^{*}(\cdot )g_{n}}$$  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 ${\displaystyle \sum \rho _{n}}$  is absolutely convergent. Nuclear operators of order 2 are called Hilbert–Schmidt operators.

Relation to trace-class operators

With additional steps, a trace may be defined for such operators when ${\displaystyle A=B.}$ 

Generalizations

More generally, an operator from a locally convex topological vector space ${\displaystyle A}$  to a Banach space ${\displaystyle B}$  is called nuclear if it satisfies the condition above with all ${\displaystyle f_{n}^{*}}$  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 ${\displaystyle f:A\to B}$  in a monoidal category is called thick if it can be written as a composition $${\displaystyle A\cong I\otimes A{\stackrel {t\otimes \operatorname {id} _{A}}{\longrightarrow }}B\otimes C\otimes A{\stackrel {\operatorname {id} _{B}\otimes s}{\longrightarrow }}B\otimes I\cong B}$$  for an appropriate object ${\displaystyle C}$  and maps ${\displaystyle t:I\to B\otimes C,s:C\otimes A\to I,}$  where ${\displaystyle I}$  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

Suppose that ${\displaystyle f:H_{1}\to H_{2}}$  and ${\displaystyle g:H_{2}\to H_{3}}$  are Hilbert-Schmidt operators between Hilbert spaces. Then the composition ${\displaystyle g\circ f:H_{1}\to H_{2}}$  is a nuclear operator.^[3]

References

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. MR 0075539
• A. Grothendieck (1956), La theorie de Fredholm, Bull. Soc. Math. France, 84:319–384. MR 0088665
• A. Hinrichs and A. Pietsch (2010), p-nuclear operators in the sense of Grothendieck, Mathematische Nachrichen 283: 232–261. doi:10.1002/mana.200910128. MR 2604120
• 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