# Xavier Mary

Laboratoire Modal'X
Université Paris Nanterre

Bat. G, Bureau E10
200 avenue de la République,
92000 Nanterre, France

E-mail: xavier.mary at parisnanterre.fr

## Research Interests

• Semigroups : structure theory, Green's relations, S-acts

• Generalized Inverses in semigroups and rings

• Hilbert subspaces and Reproducing Kernel Hilbert Spaces, Subdualities, Krein Spaces

• Determinantal processes and sampling

## In preparation

• Centralizer's applications to the inverse along an element, with H. Zhu, J. Chen and P. Patricio.

• On Hartwig-Nambooripad orders, with A. Guterman and P. Shteyner.

• Partial orders based on inverses along elements, with A. Guterman and P. Shteyner.

• Determinantal sampling designs, with V. Loonis

• On a few statistical applications of determinantal point processes with R. Bardenet, F. Lavancier and A. Vasseur

• with A. Chambaz and E. Joly, to appear as a book chapter in Targeted Learning in Data Science, by S. Rose and M. J. van der Laan (Springer, 2017).

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

We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommuting weak inverses, in semigroups and general rings. In particular, we provide various isomorphisms between the (one-sided, commuting and bicommuting) weak inverses of ab and those of ba.

We study semigroups that behave nicely with respect to a distinguished subset of idempotents E, both in terms of the extended Green’s relations KE and as unary semigroups. New structure theorems are given, notably in the case of central idempotents. Finally, the decomposition theorems are applied to the study of regular semigroups with particular generalized inverses.

[14] X. Mary and P. Patricio, The group inverse of a product. Linear and Multilinear Algebra 64 (2016), no. 9, 1776--1784.

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

In this paper, we provide equivalent conditions for the two-sided reverse order law for the group inverse: (ab)# = b#a# and (ba)# = a#b#, in semigroups and rings. Moreover, we prove that under finiteness conditions, these conditions are also equivalent with the one-sided reverse order law (ab)# = b#a# .

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation H, or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.

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.

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.

(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.

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 $Aa = Aa^2$.

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.  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.

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

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).

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.