On the nuclearity of integral operators

Let X be a nonempty measurable subset of and consider the restriction of the usual Lebesgue measure σ of to X. Under the assumption that the intersection of X with every open ball of has positive measure, we find necessary and sufficient conditions on a L²(X)-positive definite kernel in order that the associated integral operator be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of .      

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

Sobolev spaces with only trivial isometries, II

In this article, we obtain a canonical form for surjective linear isometries provided U is an open, bounded, connected, domain with Lipschitz boundary, and . We will show there exists |c| = 1 and mapping τ that is a composition of a translation and a sign-changing permutation of coordinates such that Tf = cf(τ). As a corollary, if , all surjective isometries have this trivial form by the Sobolev Imbedding Theorem.      

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

Orbits of Positive Operators from a Differentiable Viewpoint

Let be a unital C^*-algebra and G the group of units of . A geometrical study of the action of G over the set ⁺ of all positive elements of is presented. The orbits of elements with closed range by this action are provided with a structure of differentiable homogeneous space with a natural connection. The orbits are partitioned in 'components' which also have a rich geometrical structure.     

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by

Multivariable moment problems

In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C^*-algebras generated by universal row contractions associated with , the free semigroup with n generators. This class of C*-algebras includes the Cuntz-Toeplitz algebra (resp. ) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex free semigroup algebras such as ,and, more generally, the quotient algebra , where J is an arbitrary two-sided ideal of . All these results are extended to the generalized Cuntz algebra , where G_i⁺ are the positive cones ofdiscrete subgroups G_i⁺ of the real line . Moreover, we characterize the orbits of Hilbert modules over the quotient algebra , where J is an arbitrary two-sided ideal ofthe free semigroup algebra .     

Blow-up criterion for 2-D Boussinesq equations in bounded domain

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U _t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control and by and . Furthermore, can control by using vorticity transportation equations. At last, can control . Thus, we can find a blow-up criterion in the form of .      

Gelfand-Hille type theorems in ordered Banach algebras

We consider the Gelfand-Hille Theorems, specifically conditions under which an element in an ordered Banach algebra (A,C) with spectrum {1} is the identity of the algebra. In particular we show that for , where C is a closed normal algebra cone, if and x is doubly Abel bounded then x = 1. Furthermore in the case where and C is a closed proper algebra cone, then x = 1 if and only if x^L is Abel bounded and for some .      

A bilinear oscillatory integral along parabolas

We establish an norm estimate for a bilinear oscillatory integral operator along parabolas incorporating oscillatory factors . The first author was partially supported by NSF grant of China grant 10671079. The second author was supported by NSF grant DMS-0456976.     

Boundedness and compactness of positive integral operators on cones of homogeneous groups

Necessary and sufficient conditions on a weight function v guaranteeing the boundedness/compactness of integral operators with positive kernels defined on cones of homogeneous groups from L^p to L_v^q are established, where or . Behavior of singular numbers for these operators is also studied. The work was partially supported by the INTAS grant No. 05-1000008-8157 and the Georgian National Foundation Grant No. GNSF/ST06/3-010.     

Lp Contraction Semigroups for Vector Valued Functions

Let be a contraction semigroup on the space of vector valued functions ( is a Hilbert space). In order to study the extension of to a contaction semigroup on , Shigekawa [Sh] studied recently the domination property where is a symmetric sub-Markovian semigroup on . He gives in the setting of square field operators sufficient conditions for the above inequality. The aim of the present paper is to show that the methods of [12] and [13] can be applied in the present setting and provide two ways for the extension of to We give necessary and sufficient conditions in terms of sesquilinear forms for the contractivity property as well as for the above domination property in a more general situation.     

The Torsion Theory Generated by <Emphasis Type="Italic">M</Emphasis>-Small Modules

Let M be a right R-module, the class of all M-small modules, and P a projective cover of M in [M]. We consider the torsion theories = ( ), = ( ), and = ( ) in [M], where is the torsion theory generated by is the torsion theory cogenerated by , and is the dual Lambek torsion theory. We study some conditions for to be cohereditary, stable, or split, and prove that Rej(M, ) = M = (= = ) = Gen_M(P) .2000 Mathematics Subject Classification: 16S90     

Some properties of essential spectra of a positive operator, II

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ_ew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ_ef(T), then for every a ≠ 0, where T₋₁ is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if , where , then an operator on D_i is band irreducible.      

Second-Order Efficient Test for Inhomogeneous Poisson Processes

We observe a realization X ⁽ⁿ⁾ of a Poisson process on the set with intensity function depending on the unknown real parameter . Based on X ⁽ⁿ⁾ we test simple null hypothesis against one sided alternative for given . We improve the level of the well-known locally asymptotically uniformly most powerful (LAUMP) test by using the Edgeworth type expansion for stochastic integral. We show that the improved test is second-order efficient under certain regularity conditions.      

Tight Gaussian 4-Designs

A Gaussian t-design is defined as a finite set X in the Euclidean space ℝⁿ satisfying the condition: for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝⁿ, then . We call X a tight Gaussian 2e-design in ℝⁿ if holds. In this paper we study tight Gaussian 2e-designs in ℝⁿ. In particular, we classify tight Gaussian 4-designs in ℝⁿ with constant weight or with weight . Moreover we classify tight Gaussian 4-designs in ℝⁿ on 2 concentric spheres (with arbitrary weight functions).     

Ratios of Norms for Polynomials and Connected n-width Problems

Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:

The Essential Norm of Hankel Operators on the Weighted Bergman Spaces with Exponential Type Weights

Let denote the closed subspace of consisting of analytic functions in the unit disc . For certain class of subharmonic functions and , it is shown that the essential norm of Hankel operator is comparable to the distance norm from H_f to compact Hankel operators.     

Tauberian conditions for almost convergence

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in (or in ) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of (or ), where . We also show that our results can be used to derive Fatou’s theorem.