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 .      

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.      

Regular orbits and positive directions

Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14] by F. León and V. Müller. In this paper we study positive supercyclicity for operators A of the form , with , defined on . We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central throughout the paper.      

Compactness of Bochner representable operators on Orlicz spaces

We study compactness properties of linear operators from an Orlicz space L^Φ provided with a natural mixed topology to a Banach space (X, || · ||_X). We derive that every Bochner representable operator is -compact. In particular, it is shown that every Bochner representable operator is (τ(L^∞, L¹), || · ||_X)-compact.      

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

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

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.     

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.      

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

Sampling Theory Associated with a Symmetric Operator with Compact Resolvent and De Branges Spaces

Let be a symmetric operator with compact resolvent defined in a Hilbert space For any fixed we consider an entire function K_a which involves the resolvent of Associated with K_a we obtain, by duality in a Hilbert space of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of regardless of the anti-linear mapping which has as its range space. There exists also a sampling formula allowing to recover any function in from its samples at the sequence of eigenvalues of This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.     

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

On Confining Potentials and Essential Self-Adjointness for Schrödinger Operators on Bounded Domains in {\mathbb{R}}^n

Let Ω be a bounded domain in with C²-smooth boundary, , of co-dimension 1, and let be a Schr?dinger operator on Ω with potential . We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on . As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition where . More precisely, we show that if, as x approaches ,

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 .     

Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the p(x)-Laplacian

We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the p(x)-Laplacian of the form

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Boundedness of Marcinkiewicz Integral

A weighted norm inequality for the Marcinkiewicz integral operator is proved when belongs to . We also give the weighted L^p-boundedness for a class of Marcinkiewicz integral operators with rough kernels and related to the Littlewood-Paley -function and the area integral S, respectively.     

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

In the first part of these notes, we deal with first order Hamiltonian systems in the form where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian is required to be invariant with respect to the action of a group of isometries where is skew-symmetric and JA  = AJ. A standing wave is a solution having the form for some and such that . Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves. In the second part we consider the nonlinear Schr?dinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability. Received: January 2008     

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 Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

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.      

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.