Schur product techniques for commuting multivariable weighted shifts

In this paper we study the hyponormality and subnormality of 2-variable weighted shifts using the Schur product techniques in matrices. As applications, we generalize the result in [R. Curto, J. Yoon, Jointly hyponormal pairs of subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006) 5135-5159, Theorem 5.2] and give a non-trivial, large class satisfying the Curto-Muhly-Xia conjecture [R. Curto, P. Muhly, J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35 (1988) 1-22] for 2-variable weighted shifts. Further, we give a complete characterization of hyponormality and subnormality in the class of flat, contractive, 2-variable weighted shifts T≡(T₁,T₂) with the condition that the norm of the 0th horizontal 1-variable weighted shift of T is a given constant.     

One-Step Extensions of Subnormal 2-Variable Weighted Shifts

We study one-step extensions of 2-variable weighted shifts. We provide necessary and sufficient conditions for the subnormality of such extensions, by using backward extensions, disintegration of measures, and k-hyponormality techniques from the theory of 2-variable weighted shifts. We apply our results to solve an interpolation problem for measures on ${\mathbb{R}_+^2}$ .     

Prolongement d'un courant négatif plurisousharmonique avec condition sur les tranches

In this Note, we prove a theorem on the extension of a negative (or positive) plurisubharmonic current T (i.e. such that $d{d}^{c}T?0$) with condition on the slices with respect to some coordinates. This theorem generalizes a result proved by El Mir–Ben Messaoud relative to d-closed positive currents with a condition on slices. The method consists first of proving a Chern–Levine–Nirenberg inequality for a positive (or negative) psh current, which is a generalization of results obtained by Bedford–Taylor, Demailly and Sibony for d-closed positive currents. Also we prove an Oka type inequality for positive psh currents, thereby generalizing former results by Ben Messaoud–El Mir concerning positive currents with a negative $d{d}^c$. To cite this article: M. Toujani, H. Ben Messaoud, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

La forme trace d'une algèbre simple centrale de degré 4

Let k be a field of characteristic different from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple k-algebra A of degree 4 decomposes in the Witt group of k as the sum of a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ which are uniquely determined by A. The form $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A{?}_{k}A$, and $q_4$ is hyperbolic if and only if A is a symbol algebra. To cite this article: M. Rost et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Normal,unipotent subgroup schemes of reductive groups

Let G be a reductive group over a field k of characteristic p. Let $k^{\mathrm{sep}}$ be a separable closure of k. If $p\ne 2$, there exists a linear representation of G that is faithful and semisimple; moreover, any unipotent, normal subgroup scheme of G is trivial. For $p=2$, these two properties hold if and only if $G_{k^{\mathrm{sep}}}$ has no direct factor that is isomorphic to ${\mathbf{SO}}_{2n+1}$ for some $n?1$. To cite this article: A. Vasiu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Résolutions flasques des groupes réductifs connexes

A connected reductive group G over a (characteristic zero) field k may be written as a quotient $H/S$, where the k-group H is an extension of a quasitrivial torus by a simply connected group, and S is a flasque k-torus, central in H. Such presentations $G=H/S$ lead to a simplified approach to the Galois cohomology of G and related objects, such as the Brauer group of a smooth compactification of G. When k is a number field, one also recovers known formulas, in terms of S, for the quotient of the group $G(k)$ of rational points by R-equivalence, and for the Abelian groups which measure the lack of weak approximation and the failure of the Hasse principle for principal homogeneous spaces. To cite this article: J.-L. Colliot-Thélène, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On k-hyperexpansive operators

This paper considers the k-hyperexpansive Hilbert space operators T (those satisfying , 1?n?k) and the k-expansive operators (those satisfying the above inequality merely for n=k). It is known that if T is k-hyperexpansive then so is any power of T; we prove the analogous result for T assumed merely k-expansive. Turning to weighted shift operators, we give a characterization of k-expansive weighted shifts, and produce examples showing the k-expansive classes are distinct. For a weighted shift W that is k-expansive for all k (that is, completely hyperexpansive) we obtain results for k-hyperexpansivity of back step extensions of W. In addition, we discuss the completely hyperexpansive completion problem which is parallel to Stampfli's subnormal completion problem.     

When does the k-hyponormality of a 2-variable weighted shift become subnormality?

In this article we construct a sequence of nontrivial classes of 2-variable weighted shifts such that the k-hyponormality of an arbitrary power of a member W(α,β) from Gk is equivalent to its subnormality.     

Counting cubic extensions with given quadratic resolvent

Given a number field k and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of k whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha })$, for an explicit $\alpha <1$.