Convolution type operators on cones and their finite sections

This paper is concerned with finite sections of convolution type operators defined on cones, whose symbol is the Fourier transform of an integrable function on ?². The algebra of these finite sections satisfies a set of axioms (standard model) that ensures some asymptotic properties like the convergence of the condition numbers, singular values, ε‐pseudospectrum and also gives a relation between the singular values of an approximation sequence and the kernel dimensions of a set of associated operators. This approach furnishes a method to determine whether a Fredholm convolution operator on a cone is invertible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 < ∞.     

Weighted Composition Operators on H 2 and Applications

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces. Submitted: January 30, 2007. Revised: June 19, 2007. Accepted: July 11, 2007.     

The Endomorphism Kernel Property in Finite Distributive Lattices and de Morgan Algebras

Abstract

An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.     

On multicyclic operators and the Vasjunin-Nikol'ski206-1206-1206-1discotheca

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ? R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A ² R+?)^?. It is shown that if the set {λ ∈ ?: dim ker (λ-A)^* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skiï. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If M_λ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then M_λ is cyclic, but disc M_λ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

Szego-Type Limit Theorems for Generalized Discrete Convolution Operators

We study the asymptotic behavior of the averaged f-trace of a truncated generalized multidimensional discrete convolution operator as the truncation domain expands. By definition, the averaged f-trace of a finite-dimensional operator A is equal to , where n is the dimension of the space in which the operator A acts, the set of numbers γ_k, k = 1,..., n, is the complete collection of eigenvalues of the operator A, counting multiplicity; a generalized discrete convolution is an operator from the closure of the algebra generated by discrete convolution operators and by operators of multiplication by functions admitting a continuous continuation onto the sphere at infinity.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 265–277.Original Russian Text Copyright © 2005 by I. B. Simonenko.     

On some algebra of continuous linear operators

We present a criterion for an operator on L _p to belong to the set I _p of all sums of integral operators on L _p and multiplication operators by functions in L _∞. We describe the closure of I _p in the operator norm. We prove that the set L _p,1 of all sums of multiplication operators and operators on L _p mapping the unit ball of L _p into compact subsets of L ₁ is a Banach algebra.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Integral Operators with Periodic Kernels in Spaces of Integrable Functions

We consider integral operators with periodic kernels acting from Lp(ℝn) to Lq(ℝn). We obtain sufficient conditions for boundedness of that operators. Moreover we obtain compactness conditions for the product of the integral operator with periodic kernel and the operator of multiplication by an essentially bounded function.     

Formal Expansion of Harish-Chandra Homomorphism

Abstract

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G and let 𝔥 be a Cartan subalgebra of 𝔤. Let Â(𝔤) and Â(𝔥) denote the algebra of differential operators with formal power series coefficients on 𝔤 and 𝔥 respectively. We construct a subalgebra A _𝔤 of Â(𝔤) containing all the pull-backs of the differential operators in G attached to any element x in 𝔤. We also consider the projection P: A _𝔤 → Â _𝔥. Then, we calculate explicity the pull-back of the differential operator in G attached to an element h in 𝔥 modulo Ker P.     

C*-algebras of singular integral operators in domains with oscillating conical singularities

 For a domain with singular points on the boundary, we consider a C ^* -algebra of operators acting in the weighted space . It is generated by the operators of multiplication by continuous functions on and the operators where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for . When combined with the local principle, this leads to describing the Fredholm operators in . Received: 21 December 2000     

The Lower Socle and Finite Rank Elementary Operators

Abstract

We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a _k , b _k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ? a ₁ xb ₁ + ? + a _n xb _n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.     

The PDE-preserving operators on nuclearly entire functions of bounded type

The PDE-preserving operators O on the space of nuclearly entire functions of bounded type H_Nb(E) on a Banach space E are characterized. An operator is PDE-preserving when it preserves homogenous solutions to homogeneous convolution equations. We establish a one to one correspondence between O and a set Σ of sequences of entire functionals, i.e. exponential type functions. In this way, algebraic structures on Σ, such as ring structures, can be carried over to O and vice versa. In particular, it follows that O is a non-commutative ring (algebra) with unity with respect to composition and the convolution operators form a commutative subring (subalgebra). We discuss range and kernel properties, for the operators in O, and characterize the projectors (onto polynomial spaces) in O by determining the corresponding elements in Σ. This revised version was published online in June 2006 with corrections to the Cover Date.     

Representation of Simple Symmetric Operators with Deficiency Indices (1, 1) in de Branges Space

Recently it has been shown that any regular simple symmetric operator with deficiency indices (1, 1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges–Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by the independent variable in a de Branges space of entire functions. This leads to simple new results on the spectra of such symmetric operators, on when multiplication by z is densely defined in de Branges–Rovnyak spaces in the upper half plane, and to sufficient conditions for there to be an isometry from a given subspace of L² (\mathbbR, dn){L^2 (\mathbb{R}, d\nu)} onto a de Branges space of entire functions which acts as multiplication by a measurable function.     

An algebraic multigrid method for finite element discretizations with edge elements

This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H₀(curl,Ω). The finite element spaces are generated by Nédélec's edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl‐operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Irreducible representations over the diamond Lie algebra

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra 𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra 𝔢(2).     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.