On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space which can roughly be described as follows: (1) If is an open subset of and all spectral subspaces for A corresponding to compact subsets of have finite rank of negativity, the same is true for a selfadjoint operator B in for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood of such that the restriction of A to a spectral subspace for A corresponding to is a nonnegative operator in is preserved under relative perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.     

Estimates of Marcinkiewicz Integrals with Bounded Homogeneous Kernels of Degree Zero

Under the cancellation property and a certain Dini-type condition on kernels, we prove that Marcinkiewicz integrals with kernels which are homogeneous functions of degree zero, are bounded from to , from to , and from to for .     

Complete Positivity of Elementary Operators on C*-Algebras

Let be a C*-algebra. We obtain some conditions that are equivalent to the statement that every n-positive elementary operator on is completely positive.     

Spectral triangles of Schrödinger operators with complex potentials

Consider the Schrödinger operator with a complex-valued potential v of period Let and be the eigenvalues of L that are close to respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let be the diameter of the spectral triangle with vertices We prove the following statement: If then v(x) is a Gevrey function, and moreover      

The normal subgroup structure of the extended Hecke groups

We consider the extended Hecke groups generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups . Then, we determine the abstract group structure of the commutator subgroups , the even subgroup , and the power subgroups of the extended Hecke groups . Also, finally, we give some relations between them.     

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

Transformations of Grassmannians preserving the class of base subsets

Let be a finite-dimensional projective space and be the Grassmannian consisting of all k-dimensional subspaces of . In the paper we show that transformations of sending base subsets to base subsets are induced by collineations of to itself or to the dual projective space . This statement generalizes the main result of the authors paper [19].     

Equations arising from Jordan *-derivation pairs

Summary. We study certain functional equations derived from the definition of a Jordan *-derivation pair. More precisely, if A is a complex *-algebra and M is a bimodule over A, having the structure of a complex vector space compatible with the structure of A, such that implies m = 0 and implies m = 0 and if are unknown additive mappings satisfying then E and F can be represented by double centralizers. The obtained result implies that one of the equations in the definition of a Jordan *-derivation pair is redundant. Furthermore, a remark on the extension of this result to unknown additive mappings such that is given in a special case.     

C*-Modular Vector States

Let be a C*-algebra and X a Hilbert C* -module. If is a projection, let be the p-sphere of X. For φ a state of with support p in and consider the modular vector state φ_x of given by The spheres provide fibrations

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.     

A characterization of the periods of periodic points of 1-norm nonexpansive maps

In this paper the set of minimal periods of periodic points of 1-norm nonexpansive maps is studied. This set is denoted by R(n). The main goal is to present a characterization of R(n) by arithmetical and combinatorial constraints. More precisely, it is shown that , where denotes the set of periods of restricted admissible arrays on 2n symbols. The important point of this equality is that is determined by arithmetical and combinatorial constraints only, and that it can be computed in finite time. By using this equality the set R(n) is computed for . Furthermore it is shown that the largest element of R(n) satisfies:      

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Let E be a non empty set, let P : = E × E, := {x × E|x ∈ E}, := {E × x|x ∈ E}, and := {C ∈ 2 ^P |∀X ∈ : |C ∩ X| = 1} and let . Then the quadruple resp. is called chain structure resp. maximal chain structure. We consider the maximal chain structure as an envelope of the chain structure . Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms , , , related to a maximal chain structure . The set of all chains can be turned in a group such that the subgroup of generated by the left-, by the right-translations and by ι the inverse map of is isomorphic to (cf. (2.14)).     

Stability of the Cauchy congruence on restricted domain

Let S be a real interval with , and be a function satisfying We show that if h is Lebesgue or Baire measurable, then there exists such that That result is motivated by a question of E. Manstaviius. Received: 11 February 2003     

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

We consider the three-dimensional Schrödinger operators and where , A is a magnetic potential generating a constant magnetic field of strength , and where decays fast enough at infinity. Then, A. Pushnitskis representation of the spectral shift function (SSF) for the pair of operators is well defined for energies We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Communicated by Bernard HelfferSubmitted 23/09/03, accepted 15/01/04     

On quadratic differences that depend on the product of arguments - revisited

Summary. Let be a field of real or complex numbers and denote the set of nonzero elements of . Let be an abelian group. In this paper, we solve the functional equation f ₁ (x + y) + f ₂ (x - y) = f ₃ (x) + f ₄ (y) + g(xy) by modifying the domain of the unknown functions f ₃, f ₄, and g from to and using a method different from [3]. Using this result, we determine all functions f defined on and taking values on such that the difference f(x + y) + f (x - y) - 2 f(x) - 2 f(y) depends only on the product xy for all x and y in      

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.