# Hilbertian subspaces, subdualities and applications

Abstract : Functions of two variables appearing in integral transforms (Bergman, Segal, Carleman), or more generally kernels in the sense
of Laurent Schwartz - defined as weakly continuous linear mappings between the dual of a locally convex vector space
and itself - have been investigated for half a century, particularly in the field of distributions, differential equations and in the probability field with the study
of Gaussian measures or Gaussian processes.

The study of these objects may take various forms, but in case of positive kernels, the study of the properties of the image space initiated by Moore, Bergman
and Aronzjan leads to a crucial result:
the range of the kernel can be endowed with a natural
scalar product that makes it a prehilbertian space and its completion belongs ( under some weak additional topological conditions on the locally convex space)
to the locally convex space. Moreover, this injection is continuous. Positive kernels then seem to be deeply related to some particular Hilbert spaces and
our will in this thesis is to study the other kernels. What can we say if the kernel is neither positive, nor Hermitian ?

To do this we actually follow a second path and study directly spaces rather than kernels. Considering Hilbert spaces, some mathematicians have been interested
in a particular subset of the set of Hilbert spaces, those Hilbert spaces that are continuously included in a common
locally convex vector space. The relative theory is known as the theory of Hilbertian subspaces and is closely
investigated in the first chapter. Its main result is that surprisingly the notions of Hilbertian subspaces and positive kernels are equivalent, which is generally summarized as follows:

there exists a bijective correspondence between positive kernels and Hilbertian subspaces''.

The main difference with the existing theory in the first chapter is the use of dual systems and bilinear forms and one of its consequence is the emergence of some loss of symmetry that will lead to our general theory of subdualities.

In the second chapter we study the existing theory of Hermitian (or Krein) subspaces which are indefinite inner product spaces.
These spaces actually generalize the previous notion of Hilbertian subspaces and their study is a first step to the greater generalization of chapter three.
These spaces are deeply connected to Hermitian kernels but interestingly enough the previous fundamental equivalence is lost. Then we focus on the differences between this theory and the Hilbertian one for these differences will of course remain when dealing with subdualities.

In the third chapter we present a new theory of a dual system of vector spaces called subdualities which treat the previous chapters as particular cases.
A topological definition of subdualities is as follows:
a duality $(E,F)$ is a subduality of the dual system $(\cE,\cF)$ if and only if both $E$ and $F$ are weakly continuously embedded in $\cE$.
It appears that we can associate a unique kernel (in the sense of L. Schwarz) with any subduality,
whose image is dense in the subduality.
The study of the image of a subduality by a weakly continuous linear operator, makes it possible to define a vector space structure upon the set of subdualities, but given a certain equivalence relation.
A canonical representative entirely defined by the kernel is then given, which enables us to state a bijection theorem between canonical subdualities and kernels.

Finally a fourth chapter is dedicated to applications. We first study the link between Hilbertian subspaces and Gaussian measures and try to extend the theory to Krein subspaces and subdualities. Then we study some particular operators: operators in evaluation subdualities (subdualities of $\KK^(\Omega)$)
and differential operators.
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Document type :
Theses
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Cited literature [47 references]

https://tel.archives-ouvertes.fr/tel-00007004
Contributor : Xavier Mary <>
Submitted on : Thursday, September 30, 2004 - 11:56:08 AM
Last modification on : Tuesday, February 5, 2019 - 11:44:17 AM
Long-term archiving on: : Thursday, September 13, 2012 - 12:05:18 PM

### Identifiers

• HAL Id : tel-00007004, version 1

### Citation

Xavier Mary. Hilbertian subspaces, subdualities and applications. Mathematics [math]. INSA de Rouen, 2003. English. ⟨tel-00007004⟩

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