**Laboratoire
Modal'XUniversité Paris-Ouest Nanterre-La Défense
**

Bat. G, Bureau E10

200 avenue de la République,

92000 Nanterre, France

Semigroups, Generalized Inverses.

Hilbert subspaces and Reproducing Kernel Hilbert Spaces, subdualities, Krein Spaces.

(*Papers
are avaible on request. Please send an e-mail).*

X. Mary. Classes of semigroups modulo Green's relation H. (To appear in Semigroup Forum)

X. Mary. Reverse order law for the group inverse in semigroups and rings (To appear in Communications in Algebra)

[11] X. Mary, Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras. Eur. J. Pure Appl. Math. 5 (2012), 160-173 .

*Using
the recent notion of inverse along an element in a semigroup, and the
natural partial order on idempotents, we study bicommuting
generalized inverses and define a new inverse called natural inverse,
that generalizes the Drazin inverse in a semigroup, but also the
Koliha-Drazin inverse in a ring. In this setting we get a core
decomposition similar to the nilpotent, Kato or Mbekhta
decompositions. In Banach and Operator algebras, we show that the
study of the spectrum is not sufficient, and use ideas from local
spectral theory to study this new inverse. *

[10] X. Mary and P. Patricio. Generalized inverses modulo H in semigroups and rings. Linear and Multilinear Algebra 61(8) (2013), 1130-1135 .

*The definition of the
inverse along an element was very recently introduced, and it
contains known generalized inverses such as the group, Drazin and
Moore–Penrose inverses. In this article, we first prove a
simple existence criterion for this inverse in a semigroup, and then
relate the existence of such an inverse in a ring to the ring units.
*

[9] X. Mary and P. Patricio. The inverse along a lower triangular matrix. Appl. Math. Comput 219 (3)(2012) 886-891.

*In
this paper, we investigate the recently defined notion of inverse
along an element in the context of matrices over a ring. Precisely,
we study the inverse of a matrix along a lower triangular matrix,
under some conditions. *

[8] X. Mary, On generalized inverses and Green's relations. Linear Algebra Appl. 434 (2011), no. 8, 1836-1844.

(Reprint: Linear Algebra Appl. 438 (2013), no. 4, 1532–1540.)

*We
study generalized inverses on semigroups by means of Green’s
relations. We first define the notion of inverse along an element and
study its properties. Then we show that the classical generalized
inverses (group inverse, Drazin inverse and Moore–Penrose
inverse) belong to this class. *

[7] X. Mary, Moore-Penrose inverse in Krein spaces. Integral Equations Operator Theory 60 (2008), no. 3, 419-433.

*We
discuss the notion of Moore-Penrose inverse in Kreĭn spaces for
both bounded and unbounded operators. Conditions for the existence of
a Moore-Penrose inverse are given. We then investigate its relation
with adjoint operators, and study the involutive Banach algebra .
Finally applications to the Schur complement are given. *

*We
prove that the converse of Theorem 9 in “On generalized
inverses in C^*-algebras” by Harte and Mbekhta (Studia Math.
103 (1992)) is indeed true : a element $a$ of a unitary ring is group
invertible if and only if n $aA = a^2A$ and $Aa = Aa^2$.*

[5] S. Canu, C. S. Ong et X. Mary, Splines with Non-Positive Kernels. In Proceedings of the 5th ISAAC Congress , World Scientific, (2006), 1-10.

*Non
parametric regression methods can be presented in two main clusters.
The one of smoothing splines methods requiring positive kernels and
the other one known as Nonparametric Kernel Regression allowing the
use of non positive kernels such as the Epanechnikov kernel. We
propose a generalization of the smoothing spline method to include
kernels which are still symmetric but not positive semi definite
(they are called indefinite). The general relationship between
smoothing splines, Reproducing Kernel Hilbert Spaces (RKHS) and
positive kernels no longer exists with indefinite kernels. Instead
the splines are associated with functional spaces called Reproducing
Kernel Krein Spaces (RKKS) endowed with an indefinite inner product
and thus not directly associated with a norm. Smoothing splines in
RKKS have many of the interesting properties of splines in RKHS, such
as orthogonality, projection and representer theorem. % and
generalization bounds. We show that smoothing splines can be defined
in RKKS as the regularized solution of the interpolation problem.
Since no norm is available in an RKKS, Tikhonov regularization cannot
be defined. Instead, we propose the use of conjugate gradient type
iterative methods, with early stopping as a regularization mechanism.
Several iterative algorithms are collected which can be used to solve
the optimization problems associated with learning in indefinite
spaces. Some preliminary experiments with indefinite kernels for
spline smoothing reveal the computational efficiency of this
approach.*

*We
present a new theory of dual systems of vector spaces that extends
the existing notions of reproducing kernel Hilbert spaces and Hilbert
subspaces. In this theory, kernels (understood as operators rather
than kernel functions) need not be positive or self-adjoint. These
dual systems, called subdualities, enjoy many properties similar to
those of Hilbert subspaces and include the notions of Hilbert
subspaces or Kreîn subspaces as particular cases. Some
applications to Green operators or invariant subspaces are given.*

[3] A. Rakotomamonjy, X. Mary et S. Canu, Non-parametric regression with wavelet kernels. Appl. Stoch. Models Bus. Ind. 21 (2005), no. 2, 153-163.

*This
paper introduces a method to construct a reproducing wavelet kernel
Hilbert spaces for non-parametric regression estimation when the
sampling points are not equally spaced. Another objective is to make
high-dimensional wavelet estimation problems tractable. It then
provides a theoretical foundation to build reproducing kernel from
operators and a practical technique to obtain reproducing kernel
Hilbert spaces spanned by a set of wavelets. A multiscale
approximation technique that aims at taking advantage of the
multiresolution structure of wavelets is also described. Examples on
toy regression and a real-world problem illustrate the effectiveness
of these wavelet kernels. Copyright © 2005 John Wiley &
Sons, Ltd *

[2] C. S. Ong, X. Mary, S. Canu et A. J. Smola, Learning with Non-Positive Kernels. In Proceedings of the 21st International Conference on Machine Learning, ACM Press, (2004), 639-646.

*In
this paper we show that many kernel methods can be adapted to deal
with indefinite kernels, that is, kernels which are not positive
semidefinite. They do not satisfy Mercer’s condition and they
induce associated functional spaces called Reproducing Kernel Kre˘ın
Spaces (RKKS), a generalization of Reproducing Kernel Hilbert Spaces
(RKHS).*

[1] X. Mary, D. De Brucq et S. Canu, Sous-dualités et noyaux (reproduisants) associés. C. R. Math. Acad. Sci. Paris - Série I : Mathématiques 336/11 (2003), 949-954.

*A
way to study non-Gaussian measures is to generalize Schwartz's theory
of Hilbertian subspaces to the non-Hilbertian case. We initiate here
a new theory based upon a given duality rather than an Hilbertian
structure. Hence, we develop the concept of subduality of a locally
convex vector space and of its associated kernel. We show in
particular that to any subduality is associated a unique kernel whose
image is dense in the subduality. The image of a subdality under a
weakly continuous linear application is also given, which enables the
definition of a vector space structure over the set of subdualities
given a equivalence relation. We then exhibit a canonical
representative. Finally, we study the particular case of subdualities
of
endowed with the product topology. *

**Université Paris Ouest :**Licence 1 économie - Mathématiques Générales (Math 1), Cours et Travaux Dirigés;

Licence 2 MIA-MIAGE – Linear and bilinear Algebra (Algèbre 2), Cours et Travaux Dirigés;

Licence 3 MIA – Analysis in Normed Vector Spaces (Analyse 3), Cours et Travaux Dirigés.

**ENSAE :**1ère année – Mathematical Fundations of Probability Theory (Cours).

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