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.
is compact if it can be written in the form[citation needed]
are (not necessarily complete) orthonormal sets.
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
A nuclear operator on a Hilbert space has the important property that a trace operation may be defined.
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
The definition of trace-class operator was extended to Banach spaces by Alexander Grothendieck in 1955.
that is, the set of all continuous or (equivalently) bounded linear functionals on
to the Banach space of continuous linear maps from
is called nuclear if it is in the image of this evaluation map.
with the sum converging in the operator norm.
With additional steps, a trace may be defined for such operators when
The trace and determinant can no longer be defined in general in Banach spaces.
-nuclear operators via Grothendieck trace theorem.
More generally, an operator from a locally convex topological vector space
is called nuclear if it satisfies the condition above with all
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.
in a monoidal category is called thick if it can be written as a composition
In the monoidal category of Banach spaces, equipped with the projective tensor product, a map is thick if and only if it is nuclear.
are Hilbert-Schmidt operators between Hilbert spaces.