Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators L _n * obtained by a modification of given operators L _n . The operators L _n;r and L _n;r * of r-times differentiable functions are also studied. We give theorems on approximation orders of functions by these operators in polynomial weight spaces.     

Schur spaces and weighted spaces of type H ∞

Abstract

We extend some results related to composition operators on H _υ(G) to arbitrary linear operators on H _υ0(G) and H _υ(G). We also give examples of rank-one operators on H _υ(G) which cannot be approximated by composition operators.     

Operator Ideals,Orthonormal Systems and Lacunary Sets

Given an orthonormal system B in some L²(u) we consider the operator ideals II_B and T_B of B-summing and B-type operators and some related ideals. We characterize by certain weak compactness properties when II_B is equal to the operator ideal II₂ of 2-summing operators. In lose that B consists of characters of a compact abelian group we characterize when II_B coincides with the operator ideal II_γ of Gauss-summing operators and when T_B coincides with the operator ideal II_p of type-2 operators. Moreover, we give a necessary and sufficient condition for Fig to contain the operator ideal II_p of p-summing operators (2 < p < ∞) and for T_B to contain the operator ideal Γ_p of p - factorable operators.     

Dirac Operator in Two Variables from the Viewpoint of Parabolic Geometry

In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model G/P of a Cartan geometry. The first operator in this sequence is locally the Dirac operator in 2 Clifford variables, D = (D ₁, D ₂), where D _i = ∑ _j e _j . ∂ _ij . It follows from the construction that this operator is invariant with respect to the action of the group G. There are 2 other G-invariant differential operators following it so that the sequence of operators is exact. We compute the local expression of these operators and show that it coincides with the operators described in [2, 5, 6] by the tools of Clifford analysis. However, it follows from our approach that the operators are invariant. The work presented here was supported by the grants GAUK 447/2004 and GA ČR 201/05/H005.     

Generalized composition and Volterra type operators between QK spaces

Abstract

In this paper, the boundedness and compactness of the generalized composition operators and the products of Volterra type operators a nd composition operators between Q_K spaces are investigated. We also give a necessary condition for multiplication operators between Q_K spaces to be bounded or compact.     

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.     

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

A new definition of canonical conformal differential operators P _k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P _k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P _k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P _k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.     

A note onC P estimates for certain kernels

For a certain class of operators defined by integral kernels, a necessary and sufficient condition is given for the belonging to the Schatten-von Neumann idealsC _P. The operators considered generalize the classical Hankel operators; the results thus extend Peller's characterization of the Hankel operators in a classC _P.     

Theorems on decomposition of operators in L 1 and their generalization to vector lattices

The Rosenthal theorem on the decomposition for operators in L ₁ is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in L ₁ is a projective component, which yields the known fact that a sum of narrow operators in L ₁ is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L ₁, in particular the Liu decomposition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 26–35, January, 2006.     

Relatively free bands

The operators h_n and i _n and their duals hmacr;_n and imacr;_n defined on the free semigroup X ⁺ for a nonempty set X by Gerhard and Petrich are proved here to be homomorphisms of onto certain subsets of X ⁺ with a suitable multiplication. From the work of the authors mentioned, these operators induce fully invariant congruences on X ⁺ corresponding to join irreducible varieties of bands if X is countably infinite. New operators on X ⁺ are defined by means of these operators which give homomorphisms in an analogous way and induce fully invariant congruences on X ⁺corresponding to all varieties of bands except for the variety of all bands, and some varieties of normal bands. The former of these was investigated by the mentioned authors and the latter must be treated differently. By means of the above operators we are able to characterize all cases of relatively free bands.     

Toeplitz operators on the Fock space: Radial component effects

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT _a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT _a such that spT _a =ess-spT _a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

On q-Parametric Szász-Mirakjan Operators

In the present paper, we introduce q-parametric Szász-Mirakjan operators. We study convergence properties of these operators S _n,q (f). We obtain inequalities for the weighted approximation error of q-Szász-Mirakjan operators. Such inequalities are valid for functions of polynomial growth and are expressed in terms of weighted moduli of continuity. We also discuss Voronovskaja-type formula for q-Szász-Mirakjan operators.     

Regularizability of some classes of mappings that are inverses of integral operators

We study the regularizability of mappings that are inverses of integral operators acting fromC(0,1) toL ₂(0,1) and possessing noninjective continuation toL ₂(0,1). We construct classes of such operators with regularizable as well as nonregularizable inverses for which the continuation of the operators toL ₂(0,1) has an infinite-dimensional kernel. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 222–229, February, 1999.     

An ordinal indexing on the space of strictly singular operators

Using the notion of S _ξ -strictly singular operators introduced by Androulakis, Dodos, Sirotkin and Troitsky, we define an ordinal index on the subspace of strictly singular operators between two separable Banach spaces. In our main result, we provide a sufficient condition implying that this index is bounded by ω ₁. In particular, we apply this result to study operators on totally incomparable spaces, hereditarily indecomposable spaces and spaces with few operators.     

Direct and converse results for operators of Baskakov-Durrmeyer type

We consder the n-th so-called operators of Baskakov-Durrmeyer type, which result from the classical Baskakov-type operators with weights p_nk, if the discrete values f(k/n) are replaced by the integral terms (n-c∫ ₀ ^∞ p _n k(t)f(t)dt. The main differences between these operators and their classical and Kantorovicvariants respectively is that they commute. We prove direct and converse theorems also for linear combinations of the operators and results of Zamansky-Sunouchi type. As an important tool for measuring the smootheness of a function we use the Ditzian-Totik modulus of smoothness and its equivalence to appropriate K-functionals. This paper is part of the author's dissertation.     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Verschiebung and Frobenius operators

Metropolis and Rota introduced the concept of the necklace ring Nr(A) of a commutative ringA. WhenA contains Q as a subring there is a natural bijection γ:Nr(A→1+tA[]. Grothendieck has introduced a ring structure on 1+tA[t] while studyingK-theoretic Chern classes. Nr(A) comes equipped with two families of operatorsF _r,V _r called the Frobenius and Verschiebung operators. Mathematicians studying formal group laws have introduced two families of operators,F _r, andV _r on 1+tA[t]. Metropolis and Rota have not however tried to show that γ preserves, these operators. They transport the operators from Nr(A) to 1+tA[t] using γ. In our present paper we show that γ does preserve all these operators. Part of this work was done while the author was visiting the Institute of Mathematical Sciences, Madras.     

Pseudodifferential <Emphasis Type="Italic">p</Emphasis>-adic vector fields and pseudodifferentiation of a composite <Emphasis Type="Italic">p</Emphasis>-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O _p ) of balls in O _p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O _p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.     

Non Integral Regularizing Operators on Lp- Spaces

We present bounded positivity preserving operators from L_p(?) to L_q (?), for 1 < p < ∞, 1/p-1/q < 1/2, which are not integral operators.     

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L₁ is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N_r(E, F) in L_r(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.