On pointwise estimate for Gamma operators

For linear combinations of Gamma operators, if 0＜a＜2r, 1/2-1/2r≤λ≤l, or 0≤λ＜1/2-1/2r(r≥2),0＜a＜r+10＜a<(r+1)/1-λ, we obtain an equivalent theorem with ωuλρ(f ,t) instead of ωrλφ(f,t), where ωuφ(f,t) is theDitzian-Totik moduli of smoothness.     

On pointwise estimate for Gamma operators

For linear combinations of Gamma operators, if 0 \frac12r \leqslant l\leqslant 1\frac{1}{2} - \frac{1}{{2r}} \leqslant \lambda \leqslant 1 , or 0 \leqslant l < \frac12 - \frac12r(r \geqslant 2)0 \leqslant \lambda< \frac{1}{2} - \frac{1}{{2r}}(r \geqslant 2) , 0 < a < \fracr + 11 - l0< a< \frac{{r + 1}}{{1 - \lambda }} , we obtain an equivalent theorem with      

On pointwise estimates for the Littlewood-Paley operators

In a recent paper we proved pointwise estimates relating some classical maximal and singular integral operators. Here we show that inequalities essentially of the same type hold for the Littlewood-Paley operators.

     

A representation of strong Gaussian random operators

An integral representation of Gaussian random operators acting on a real separable Hilbert space is considered. In terms of this representation, boundedness conditions for Gaussian random operators are given. A special class of Gaussian random operators is studied.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 19–23, 1988.     

On the representation of the joint spectrum for a commutingn-tuple of non-normal operators

The main purpose of the present paper is to describe the Taylor joint spectra forn-tuples of double commuting hyponormal operators, and to study the representation of the joint spectra in terms of that ofn-tuples of commuting normal operators.     

Subsequence pointwise ergodic theorems for operators inL p

In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations are extended to Dunford-Schwartz operators, positive isometries, and power bounded Lamperti operators. Partially supported by NSF Grant DMS-8910947.     

A spectral integral representation for decomposable operators

For the operators commuting with a spectral measure, a Bartle type integral representation in the spirit of the one provided by the spectral theorem for normal operators is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Expansion and representation of regular operators

Yakutsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 2, pp. 192–202, March–April, 1989.     

On the rate of pointwise convergence of the Durrmeyer-type operators

For bounded or some locally bounded functions f measurable on an interval I there is estimated the rate of convergence of the Durrmeyer-type operatorsL f at those points xεIntI at which the one-sided limits f(x±0) exist. In the main theorems the Chanturiya's modulus of variation is used.     

An example of pointwise non-convergence of iterated conditional expectation operators

Given ∈, we construct a sequence , … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on , then , then , Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈. The sequence of functions also will fail to converge in the . This contrasts with the long-open conjecture that if all the come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in . J. L. King was partially supported by NSF grant DMS-9112595.     

Multiplicative representation of bilinear operators

We establish that each lattice bimorphism from the Cartesian product of two vector lattices into a universally complete vector lattice is representable as the product of two lattice homomorphisms defined on the factors. This fact makes it possible to reduce the problem to the linear case and obtain some results on representation of an order bounded disjointness preserving bilinear operator as a strongly disjoint sum of weighted shift or multiplicative operators.     

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

A REPRESENTATION FORMULA RELATED TO SCHROEDINGER OPERATORS

Schroedinger operator is a central subject in the mathematical study of quantum mechanics.Consider the SchrSdinger operator H=-Δ V on R, where Δ=d^2/dx^2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of-Δ. A natural conjecture is that an L^2 function admits a similar expansion in terms of “eigenfunctions“ of H, a perturbation of the Laplacian (see [7], Ch. XI and the notes), under certain condition on V.     

Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in Carro et al. (2005) [7] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator Mα,B associated to a Young function B and the multilinear maximal operators Mψ=M0,ψ, ψ(t)=B(t1−α/(nm))^nm/(nm−α). As an application of these estimate we obtain a direct proof of the Lp−Lq boundedness results of Mα,B for the case B(t)=t and Bk(t)=tk(1+log⁺t) when 1/q=1/p−α/n. We also give sufficient conditions on the weights involved in the boundedness results of Mα,B that generalizes those given in Moen (2009) [22] for B(t)=t. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.     

Commutation properties of anticommuting self-adjoint operators,spin representation and Dirac operators

LetA andB be anticommuting self-adjoint operators in a Hilbert space . It is proven thatiAB is essentially self-adjoint on a suitable domain and its closureC(A, B) anticommutes withA andB. LetU _s be the partial isometry associated with the self-adjoint operatorsS, i.e., the partial isometry defined by the polar decompositionS=U _S |S|. LetP _S be the orthogonal projection onto (KerS). Then the following are proven: (i) The operatorsU _A ,U _B ,U _C(A,B) ,P _A ,P _B , andP _A P _B multiplied by some constants satisfy a set of commutation relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra of the special unitary groupSU(2); (ii) There exists a Lie algebra associated with those operators; (iii) If is separable andA andB are injective, then gives a completely reducible representation of with each irreducible component being the spin representation of the Clifford algebra associated with ³; This result can be extended to the case whereA andB are not necessarily injective. Moreover, some properties ofA+B are discussed. The abstract results are applied to Dirac operators.