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. 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 operators on Hilbert space, see the article Trace class.
Compact operator
An operator on a Hilbert space is compact if it can be written in the form where 1 ≤ ≤ ∞, and and are orthonormal sets. Here are a set of real numbers, the singular values of the operator, obeying → 0 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
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. It can be shown that this trace is identical to the sum of the eigenvalues of .
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 be the dual of A, that is, the set of all continuous or bounded linear functionals on A with the usual norm. There is a canonical evaluation map . 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.
''q''-nuclear operators
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 ∑ρn 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.
Generalizations
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. 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 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.
