Integral linear operator


An integral bilinear form is a bilinear functional that belongs to the continuous dual space of, the injective tensor product of the locally convex topological vector spaces X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form.
These maps play an important role in the theory of nuclear spaces and nuclear maps.

Definition - Integral forms as the dual of the injective tensor product

Let X and Y be locally convex TVSs, let denote the projective tensor product, denote its completion, let denote the injective tensor product, and denote its completion.
Suppose that denotes the TVS-embedding of into its completion and let be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of as being identical to the continuous dual space of.
Let denote the identity map and denote its transpose, which is a continuous injection. Recall that is canonically identified with, the space of continuous bilinear maps on. In this way, the continuous dual space of can be canonically identified as a subvector space of, denoted by. The elements of are called integral forms on. The following theorem justifies the word integral.
Theorem The dual J of consists of exactly those continuous bilinear forms v on that can be represented in the form of a map
where S and T are some closed, equicontinuous subsets of and, respectively, and is a positive Radon measure on the compact set with total mass.
Furthermore, if A is an equicontinuous subset of J then the elements can be represented with fixed and running through a norm bounded subset of the space of Radon measures on.

Integral linear maps

A continuous linear map is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by. It follows that an integral map is of the form:
for suitable weakly closed and equicontinuous subsets S and T of and, respectively, and some positive Radon measure of total mass ≤ 1.
The above integral is the weak integral, so the equality holds if and only if for every,.
Given a linear map, one can define a canonical bilinear form, called the associated bilinear form on, by.
A continuous map is called integral if its associated bilinear form is an integral bilinear form. An integral map is of the form, for every and :
for suitable weakly closed and equicontinuous aubsets and of and, respectively, and some positive Radon measure of total mass.

Relation to Hilbert spaces

The following result shows that integral maps "factor through" Hilbert spaces.
Proposition: Suppose that is an integral map between locally convex TVS with Y Hausdorff and complete. There exists a Hilbert space H and two continuous linear mappings and such that.
Furthermore, every integral operator between two Hilbert spaces is nuclear. Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.

Sufficient conditions

Suppose that is a continuous linear map between locally convex TVSs.