Nuclear operators between Banach spaces explained

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

\tfrac{ 2 }{ 3 } -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

See main article: trace class operator. An operator

l L on a Hilbert space l H $$\backslash mathcal\; :\; \backslash mathcal\; \backslash to\; \backslash mathcal$$is compact if it can be written in the form$$\backslash mathcal\; =\; \backslash sum\_^N\; \backslash rho\_n\; \backslash langle\; f\_n,\; \backslash cdot\; \backslash rangle\; g\_n,$$where 1 \leq N \leq inf ty, and \{f _{1,} \ldots, f _{N\}} and \{g _{1,} \ldots, g _{N\}} are (not necessarily complete) orthonormal sets. Here \{\rho _{1,} \ldots, \rho _{N\}} is a set of real numbers, the set of singular values of the operator, obeying \rho _{n} \to 0 if N = inf ty.

The bracket

\langle ⋅ , ⋅ \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 or if

Properties

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

\{\psi _{n\}} for the Hilbert space, the trace is defined as$$\backslash operatorname\; \backslash mathcal\; =\; \backslash sum\_n\; \backslash langle\; \backslash psi\_n,\; \backslash mathcal\; \backslash psi\_n\; \backslash rangle.$$

Obviously, the sum converges absolutely, and it can be proven that the result is independent of the basis. It can be shown that this trace is identical to the sum of the eigenvalues of

l{L} (counted with multiplicity).

Nuclear operators on Banach spaces

See main article: Fredholm kernel.

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

Let

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

-nuclear operators

An operator$$\backslash mathcal\; :\; A\; \backslash to\; B$$is said to be if there exist sequences of vectors

\{g _{n\}} \in B with \Vert g _{n} \Vert \leq 1 , functionals \left\{f ^{*} _{n\right\}} \in A ^{\prime} with \Vert f ^{*} _{n} \Vert \leq 1 and complex numbers \{\rho _{n\}} withsuch that the operator may be written as$$\backslash mathcal\; =\; \backslash sum\_n\; \backslash rho\_n\; f^*\_n(\backslash cdot)\; g\_n$$with the sum converging in the operator norm.

Operators that are nuclear of order 1 are called : these are the ones for which the series

\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

A = B.

Properties

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

\tfrac{ 2 }{ 3 } -nuclear operators via Grothendieck trace theorem.

Generalizations

A to a Banach space B is called if it satisfies the condition above with all 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 . 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

f : A \to B in a monoidal category is called if it can be written as a composition$$A\; \backslash cong\; I\; \backslash otimes\; A\; \backslash stackrel\; \backslash longrightarrow\; B\; \backslash otimes\; C\; \backslash otimes\; A\; \backslash stackrel\; \backslash longrightarrow\; B\; \backslash otimes\; I\; \backslash cong\; B$$for an appropriate object C and maps t: I \to B ⊗ C, s: C ⊗ A \to I, where 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.

Examples

Suppose that

f : H _{1} \to H _{2} and g : H _{2} \to H _{3} are Hilbert-Schmidt operators between Hilbert spaces. Then the composition g \circ f : H _{1} \to H _{3} is a nuclear operator.

References

• A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,Mem. Am. Math.Soc. 16.
• A. Grothendieck (1956), La theorie de Fredholm, Bull. Soc. Math. France, 84:319–384.
• A. Hinrichs and A. Pietsch (2010), p-nuclear operators in the sense of Grothendieck, Mathematische Nachrichen 283: 232–261. .