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

An operator \mathcal L on a Hilbert space \mathcal H is compact if it can be written in the form 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 N = \infty. 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 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 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 \mathcal{L} (counted with multiplicity).

Nuclear operators on Banach spaces

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 (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 and b \in B to the linear map An operator is called if it is in the image of this evaluation map.

q-nuclear operators

An operator is said to be if there exist sequences of vectors with functionals with and complex numbers {\rho_n} 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 : 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 -nuclear operators via Grothendieck trace theorem.

Generalizations

More generally, an operator from a locally convex topological vector space 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 for an appropriate object C and maps 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 and are Hilbert-Schmidt operators between Hilbert spaces. Then the composition is a nuclear operator.