Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

Operator norms,multiplicativity factors,and C-numerical radii

Let V be a normed vector space over C, let $\text{B}$(V) denote the algebra of linear bounded operators on V, and let N be an arbitrary seminorm or norm on $\text{B}$(V). In this paper we discuss multiplicativity factors for N, i.e., constants μ>0 for which N_μ≡μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C-numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.     

The metaplectic representation,Weyl operators and spectral theory

Let X be the Grassmannian of Lagrangian subspaces of R²ⁿ and π: Θ → X the bundle of negative half-forms. We construct a canonical imbedding S(Rⁿ)_even → C^∞(Θ) which intertwines the metaplectic representation of M_p(n) on S(Rⁿ) with the induced representation of M_p(n) on C^∞(Θ). This imbedding converts the algebra of Weyl operators into an algebra of pseudodifferential operators and enables us to prove theorems about the spectral properties of Weyl operators by reducing them to standard facts about pseudodifferential operators. For instance we are able to prove a Weyl theorem on the asymptotic growth of eigenvalues with an “optimal” error estimate for such operators and an analogue of the Helton clustering theorem and the Chazarain-Duistermaat-Guillemin trace formula.     

Invariance spectrale des algèbres d'opérateurs pseudodifférentiels

We construct and study several algebras of pseudodifferential operators that are closed under holomorphic functional calculus. This leads to a better understanding of the structure of inverses of elliptic pseudodifferential operators on certain non-compact manifolds. It also leads to decay properties for the solutions of these operators. To cite this article: R. Lauter et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1095–1099.     

Classification projective des espaces d’opérateurs différentiels agissant sur les densités

One classifies the representation D_λμ of sl(m + 1, R) obtained by letting its projective embedding in the Lie algebra of vector fields of R^m, m > 1, act by Lie derivatives on the space of differential operators between densities of weight λ and μ. For each μ - λ, there is only finitely many isomorphism classes, most frequently one, in which case D_λμ is isomorphic to its graded space relative to the order of differentiation.     

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

Analyse de l'espace des phases et calcul pseudo-differential sur le groupe de Heisenberg

We establish pseudo-differential calculus on the Heisenberg group by defining an algebra of operators acting continuously on Sobolev spaces and containing the class of differential operators. Our approach puts into light microlocal directions and completes, with the Littlewood–Paley theory developed by Bahouri et al., a microlocal analysis of the Heisenberg group. To cite this article: H. Bahouri et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Multidimensional integral operators with homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients

In the space L _p (? ⁿ ), 1 < p < +∞, we consider a new class of integral operators with kernels homogeneous of degree ?n, which includes the class of operators with homogeneous SO(n)-invariant kernels; we study the Banach algebra generated by such operators with multiplicatively weakly oscillating coefficients. For operators from this algebra, we define a symbol in terms of which we formulate a Fredholm property criterion and derive a formula for calculating the index. An important stage in obtaining these results is the establishment of the relationship between the operators of the class under study and the operators of one-dimensional convolution with weakly oscillating compact coefficients.     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

有一致降指数的算子

本文讨论了有拓扑一致降指数的算子的可交换拟幂零摄动:举出反例说明有拓扑一致降指数的算子在可交换拟幂零摄动下是不稳定的;得到有拓扑一致降指数的算子和它的可交换拟幂零摄动的超值域与超核之间的关系;利用这些关系,证明了左(右)Drazin可逆算子在可交换幂零摄动下是稳定的.最后还讨论了Banach空间上B-正则类BR_i(1≤i≤13)中算子的R_i型Kato分解与R_i型超Kato分解.     

Fuzzy set approach to the utility,preference relations,and aggregation operators

Score x = (x₁, … , x_n) describing an alternative α is modelled by means of a continuous quasi-convex fuzzy quantity μ_α = μ_x, thus allowing to compare alternatives (scores) by means of fuzzy ordering (comparison) methods. Applying some defuzzification method leads to the introduction of operators acting on scores. A special stress is put on the Mean of Maxima defuzzification method allowing to introduce several averaging aggregation operators. Moreover, our approach allows to introduce weights into above mentioned aggregation, even in the non-anonymous (non-symmetric) case. Finally, Ordered Weighted Aggregation Operators (OWAO) are introduced, generalizing the standard OWA operators.     

On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C^∗-algebra elements.     

On a class of anisotropic asymptotically periodic HamiltoniansHamiltoniens anisotropes asymptotiquement périodiques

We construct a $\text{C}{}^1$-algebra $\text{C}$ proper to an anisotropic asymptotically periodic quantum system and we compute its quotient by the algebra of compact operators. We describe then the self-adjoint operators affiliated to $\text{C}$ and their essential spectrum. To cite this article: O. Rodot, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579.     

Closure operators I

Closure operators in an (E, M)-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M. Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X. Every closure operators admits an idempotent hull and a weakly hereditary core.Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotent, weakly hereditary, standard closure operators correspond to idempotent radicals (= torsion theories).     

Finite Section Method for a Banach Algebra of Convolution Type Operators on {L^p(\mathbb R)} with Symbols Generated by PC and SO

We prove necessary and sufficient conditions for the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on L^p(\mathbb R){L^p(\mathbb {R})}, 1 < p < ∞.     

Certain Jordan operator algebras and double commutant theorems

Two operations on sets of bounded operators on Hilbert space are studied: One produces Jordan algebras and the other, its inverse in a certain sense, produces self-adjoint linear spaces. The question of which Jordan algebras arise from the first of these operations is investigated. It is shown that, in many cases, R has this property if and only if the complex associative algebra R + iR is equal to its double commutant. Thus, a by-product of the study is another approach to double commutant theorems.     

Weakly tight functions, their Jordan type decomposition and total variation in effect algebras

In the present paper, we have studied envelopes of a function m defined on a subfamily E (containing 0 and 1) of an effect algebra L. The notion of a weakly tight function is introduced and its relation to tight functions is investigated; examples and counterexamples are constructed for illustration. A Jordan type decomposition theorem for a locally bounded real-valued weakly tight function m defined on E is established. The notions of total variation |m| on the subfamily E and m-atoms on a sub-effect algebra E (along with a few examples of m-atoms for null-additive as well as non null-additive functions) are introduced and studied. Finally, it is proved for a real-valued additive function m on a sub-effect algebra E that, m is non-atomic if and only if its total variation |m| is non-atomic.     

Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB ^*-algebra leads to the study of properGB ^*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB ^*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC ^*-algebra and two other representation theorems forb ^*-algebras (also calledlmc ^*-algebras), one representinga b ^*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeL ^p-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andL ^w-integral of a measurable field ofC ^*-algebras are discussed briefly.     

Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces

In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space H(k) with complete Nevanlinna-Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of H(k)⊗Cm have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems.     

Diagram algebras, Hecke algebras and decomposition numbers at roots of unity

We prove that the cell modules of the affine Temperley-Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley-Lieb algebra of type B, which provides a connection between Temperley-Lieb algebras on n and n−1 strings. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan-Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein-Zelevinsky on the “generic case” and of Jones on link invariants.