In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation.
Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).
Throughout let X,Y, and Z be topological vector spaces (TVSs) and L : X → Y be a linear operator (no assumption of continuity is made unless otherwise stated).
In a Hilbert space, positive compact linear operators, say L : H → H have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:[3] There is a sequence of positive numbers, decreasing and either finite or else converging to 0,
and a sequence of nonzero finite dimensional subspaces
; and (3) the orthogonal of the subspace spanned by
is equal to the kernel of L.[3] Let X and Y be vector spaces (no topology is needed yet) and let Bi(X, Y) be the space of all bilinear maps defined on
and going into the underlying scalar field.
be the canonical linear form on Bi(X, Y) defined by
denotes the algebraic dual of Bi(X, Y).
If Z is any other vector space then the mapping Li(X ⊗ Y; Z) → Bi(X, Y; Z) given by u ↦ u ∘ 𝜒 is an isomorphism of vector spaces.
In particular, this allows us to identify the algebraic dual of X ⊗ Y with the space of bilinear forms on X × Y.
[4] Moreover, if X and Y are locally convex topological vector spaces (TVSs) and if X ⊗ Y is given the π-topology then for every locally convex TVS Z, this map restricts to a vector space isomorphism
from the space of continuous linear mappings onto the space of continuous bilinear mappings.
[5] In particular, the continuous dual of X ⊗ Y can be canonically identified with the space B(X, Y) of continuous bilinear forms on X × Y; furthermore, under this identification the equicontinuous subsets of B(X, Y) are the same as the equicontinuous subsets of
[5] There is a canonical vector space embedding
and its elements are called nuclear operators.
and the norm on this quotient space, when transferred to elements of
Nuclear automorphisms of a Hilbert space are called trace class operators.
Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map.
Then N is a nuclear map if and only if R is a nuclear map; hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive self-adjoint operators R.[11] Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map whose absolute value is R : X → X.
The following are equivalent: Suppose that U is a convex balanced closed neighborhood of the origin in X and B is a convex balanced bounded Banach disk in Y with both X and Y locally convex spaces.
One can define the auxiliary Banach space
one obtains through composition the continuous linear map
[7] Definition: Let X and Y be Hausdorff locally convex spaces.
as U ranges over all closed convex balanced neighborhoods of the origin in X and B ranges over all bounded Banach disks in Y, is denoted by
and its elements are call nuclear mappings of X into Y.
[7] When X and Y are Banach spaces, then this new definition of nuclear mapping is consistent with the original one given for the special case where X and Y are Banach spaces.
Let X and Y be Hausdorff locally convex spaces and let
The following is a type of Hahn-Banach theorem for extending nuclear maps: Let X and Y be Hausdorff locally convex spaces and let