On the range and the kernel of the operator

Let denote the algebra of (bounded linear) operators on the separable complex Hilbert space , and let denote a norm ideal in . For , let the derivation be defined by , and let be defined by . The main result of this paper is to show that if , are contractions, then for every operator such that , then for all .

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

Ultradifferentiable functions on lines in

It is well known that a function whose restriction to every line in is real analytic must itself be real analytic. In this note we study whether this property of real analytic functions is also possessed by some other subclasses of functions. We prove that if is ultradifferentiable corresponding to a sequence on every line in some `uniform way', then is ultradifferentiable corresponding to the sequence

     

On sums and products of integers

Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

     

Irreducible plane curves with the Albanese dimension 2

Let be a plane curve given by an equation , and let be the affine plane curve given by . Let denote a cyclic covering of determined by . The number is called the Albanese dimension of . In this article, we shall give examples of with the Albanese dimension 2.

     

Helly-type theorems for hollow axis-aligned boxes

A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

     

Spectral multiplier theorem for spaces associated with some Schrödinger operators

Let be the semigroup of linear operators generated by a Schrödinger operator , where is a nonnegative polynomial. We say that is an element of if the maximal function belongs to . A criterion on functions which implies boundedness of the operators on is given.

     

Normalizers of the congruence subgroups of the Hecke group

Let cos and let be the Hecke group associated to . In this article, we show that for a prime ideal in , the congruence subgroups of are self-normalized in .

     

Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

A factorization theorem for the derivative of a function in

We show that a function is the derivative of a function in the Hardy space of the unit disk for if and only if where and . Here, can be chosen to be non-vanishing, , and . As an application, we characterize positive measures on the unit disk such that the operator is bounded from the tent space to , where .

     

The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .

     

Ample divisors on the blow up of at points

Fix integers with and ; if assume . Let be general points of the complex projective space and let be the blow up of at with exceptional divisors , . Set . Here we prove that the divisor is ample if and only if , i.e. if and only if .

     

Nonexpansive, -continuous antirepresentations have common fixed points

Let be a closed convex subset of a Banach (dual Banach) space . By we denote an antirepresentation of a semitopological semigroup as nonexpansive mappings on . Suppose that the mapping is jointly continuous when has the weak (weak*) topology and the Banach space of bounded right uniformly continuous functions on has a right invariant mean. If is weakly compact (for some the set is weakly* compact) and norm separable, then has a common fixed point in .

     

A bound for

Let denote the largest irreducible character degree of a finite group , and let be a prime. Two results are obtained. First, we show that, if is a -solvable group and if , then . Next, we restrict attention to solvable groups and show that, if and if is a Sylow -subgroup of , then .

     

Sur les -corps

Here we give new examples of fields in characteristic whose -invariant and -invariant are different: or . These fields are also -fields.

RSUM. Nous donnons ici de nouveaux exemples de corps en caractéristique dont le -invariant et le -invariant diffèrent. Plus précisément: et ou . Ces corps sont aussi des -corps.

     

The -homology class of the Euler characteristic operator is trivial

On any manifold , the de Rham operator (with respect to a complete Riemannian metric), with the grading of forms by parity of degree, gives rise by Kasparov theory to a class , which when is closed maps to the Euler characteristic in . The purpose of this note is to give a quick proof of the (perhaps unfortunate) fact that is as trivial as it could be subject to this constraint. More precisely, if is connected, lies in the image of (induced by the inclusion of a basepoint into ).

     

Toeplitz -algebras on ordered groups and their ideals of finite elements

Let be a discrete abelian group and an ordered group. Denote by the minimal quasily ordered group containing . In this paper, we show that the ideal of finite elements is exactly the kernel of the natural morphism between these two Toeplitz -algebras. When is countable, we show that if the direct sum of -groups , then .

     

Behaviour of holomorphic automorphisms on equicontinuous subsets of the space

Consider a compact Hausdorff topological space , a -triple and , the -triple of all continuous -valued functions with the pointwise operations and the norm of the supremum. Let be the group of all holomorphic automorphisms of the unit ball of that map every equicontinuous subset lying strictly inside into another such a set. The real Banach-Lie group and its Lie algebra are investigated. The identity connected component of is identified when has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case .

     

A characterization of

This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.