Integral linear operator
In mathematical analysis, an integral linear operator is a linear operator T given by integration; i.e.,
where is called an integration kernel.
More generally, 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 vector subspace of, denoted by. The elements of are called integral forms on. The following theorem justifies the word integral.
There is also a closely related formulation of the theorem above that can also be used to explain the terminology integral bilinear form: a continuous bilinear form on the product of locally convex spaces is integral if and only if there is a compact topological space equipped with a positive Radon measure and continuous linear maps and from and to the Banach space such that
i.e., the form can be realised by integrating functions on a compact space.
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
Every nuclear map is integral. An important partial converse is that every integral operator between two Hilbert spaces is nuclear.Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that,, and are all continuous linear operators. If is an integral operator then so is the composition.
If is a continuous linear operator between two normed space then is integral if and only if is integral.
Suppose that is a continuous linear map between locally convex TVSs.
If is integral then so is its transpose. Now suppose that the transpose of the continuous linear map is integral. Then is integral if the canonical injections and are TVS-embeddings.
Properties
Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If,, and are all integral linear maps then their composition is nuclear.Thus, in particular, if is an infinite-dimensional Fréchet space then a continuous linear surjection cannot be an integral operator.