Projective limits of weighted (LB)-spaces of continuous functions

Countable projective spectra of countable inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions are investigated. It is characterized when the derived projective limit functor vanishes in terms of the sequences of the weights defining the spaces. The locally convex properties of the corresponding projective limits are analyzed, too. Received: 30 January 2009     

Strong duals of projective limits of (LB)-spaces

We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called strong P-type. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.     

On projective motions of five-dimensional spaces of special form

The paper is devoted to the problem of determining of 5-dimensional pseudo-Riemannian manifolds (M, g) admitting projective motions (h-spaces). A similar problem for n-dimensional proper Riemannian and Lorentz spaces was solved by Levi-Civita, Solodovnikov, Petrov and Aminova. For pseudo-Riemannian manifolds of arbitrary signature and dimension the problem of their classification in Lie algebras and Lie groups of projective transformations, set more than a hundred years ago, is still open. In this paper five-dimensional h-spaces of the type {221} are determined using the method of skew-normal frame (Aminova) and necessary and sufficient conditions for the existence of projective motions of the same type are established.     

On the completion of (LF)-spaces

Once the existence of metrizable (LF)-spaces was discovered, the problem whether the completion of an (LF)-space is or is not an (LF)-space was answered in the negative, because no (LF)-space can be a Fréchet space. However, some (non-metrizable) (LF)-spaces are complete, e.g. the classical Köthe's strict (LF)-spaces. In this paper we will carry out a thorough study of the completeness of (LF)-spaces stressing upon the stable completion properties of (LB)-spaces. A basic tool for handling this problem is an Open Mapping Theorem for completions of (LF)-spaces, which is also proved in the present paper.This research, supported by a Fulbright-MEC fellowship, was carried out during the author's visit at the University of Maryland while on leave from the University of Extremadura. The author is grateful to Professor S. A. Saxon for very helpful conversation and, especially, for calling his attention to the problem of completions of (LF)-spaces.     

Retakh条件(M0)与弱(序列式)紧正则性

本文研究了弱(序列式)紧正则诱导极限与凸弱(序列式)紧正则诱导极限.满足Retakh条件(Mo)的(LF)-空间必为凸弱(序列式)紧正则的,但未必为弱(序列式)紧正则的.对于弱序列式完备Frechet空间的可数诱导极限,Retakh条件(Mo)蕴涵弱(序列式)紧正则性.特别地,对于Kothe(LF)-序列空间Ep(1≤p<∞),Retakh条件(Mo)等价于弱(序列式)紧正则性.对于这类空间,利用Vogt的有关结论,给出了弱(序列式)紧正则性的特征.     

The trace formula in Banach spaces

A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalues provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988, Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace formula. A natural class of asymptotically Hilbertian spaces, including some spaces that are ?₂ sums of finite-dimensional spaces, are Γ-spaces. One consequence is that the direct sum of two Γ-spaces need not be a Γ-space.     

Quantum differential systems and construction of rational structures

We consider mirror symmetry (A-side vs B-side, namely singularity side) in the framework of quantum differential systems. We focuse on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau–Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani’s results), depending on numerous combinatorial data which are rearranged after the mirror transformation.     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

On one transformation of parameter-spaces of real interpolation method

In connection with estimation of interpolation orbits and coorbits we introduce a new transformation acting in the class of all parameter-space of real interpolation K-method. We “calculate” the result of the transformation of classical parameters. It is revealed that the transformation of weighted L₁-spaces leads to Orlicz spaces.     

On isometrically universal spaces, mappings, and actions of groups

In this paper we first consider some well-known classes of separable metric spaces which are isometrically ω-saturated (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559]) and, therefore, contain isometrically universal spaces. We put some problems concerning such spaces most of which are related with the properties of the isometrically universal Urysohn space. Furthermore, using the defined notions of isometrically universal mappings and G-spaces (which are analogies of the notion of isometrically universal spaces) we introduce the notions of an isometrically ω-saturated class of mappings and an isometrically ω-saturated class of G-spaces (in which there are “many” isometrically universal elements). We prove that all results of Sections 6.1 and 7.1 of [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559] can be reformulated for isometrically ω-saturated classes of spaces and G-spaces, respectively. In particular, we prove that if D and R are isometrically ω-saturated classes of spaces, then the class of all mappings with the domain in D and range in R is an isometrically ω-saturated class of mappings and, therefore, in this class there are isometrically universal elements. As a corollary of this result we have that since the class of all mappings is isometrically ω-saturated, in this class there are isometrically universal mappings. Similarly, if G is an arbitrary separable metric group and P is an isometrically ω-saturated class of spaces, then the class of all G-spaces (X,F), where X is an element of P, is an isometrically ω-saturated class of G-spaces and, therefore, in this class there are isometrically universal elements. In particular, for any separable metric group G, in the class of all G-spaces there are isometrically universal G-spaces. We also pose some problems concerning isometrically universal mappings and G-spaces some of which concern the Urysohn space.     

Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.     

Geodesics on weighted projective spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.     

No dense metrizable Gδ-subspaces in butterfly semi-metrizable Baire spaces

A class of Baire spaces, which contains many known examples and variations thereof, is described and it is shown that no space in this class contains a dense metrizable G_δ-subspace. This gives a class of semi-metrizable spaces which are not σ-spaces. We discuss the existence of Lindelöf semi-metrizable spaces which are not σ-spaces. This is of interest since the only known examples require the use of CH.     

A predual of a predual of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript> and its applications to commutators

A predual of B_σ-spaces is investigated. A predual of a predual of B_σ-spaces is also investigated, which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation (BMO) and the singular integral operators. What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of ?_σ-spaces and B_σ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to B_σ-spaces.     

Some Questions of Heinrich on Ultrapowers of Locally Convex Spaces

In this note we treat some open problems of Heinrich on ultrapowers of locally convex spaces. In section 1 we investigate the localization of bounded sets in the full ultrapower of a locally convex space, in particular the coincidence of the full and the bounded ultrapower, mainly concentrating in the case of (DF)-spaces. In section 2 we provide a partial answer to a question of Heinrich on commutativity of strict inductive limits and ultrapowers. In section 3 we analyze the relation between some natural candidates for the notion of superreflexivity in the setting of Fréchet spaces. We give an example of a Fréchet-Schwartz space which is not the projective limit of a sequence of superreflexive Banach spaces.     

Plemelj Formulas for Rarita-Schwinger Type Operators

We introduce Plemelj formulas for Rarita-Schwinger operators defined over Lipschitz graphs in \({\mathbb{R}^{n}}\) and their corresponding surfaces on the sphere, S ⁿ and real projective spaces. We introduce the corresponding Hardy p-spaces for \({1 < p < \infty}\) . We also introduce Rarita-Schwinger analogues of the classical Szegö projection operators and Kerzman-Stein formulas.     

Energies,group-invariant kernels and numerical integration on compact manifolds

The purpose of this paper is to derive quadrature estimates on compact, homogeneous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets.     

Projective limits of MV-spaces

An MV-space is a topological space X such that there exists an MV-algebra A whose prime spectrum Spec A is homeomorphic to X. The characterization of the MV-spaces is an important open problem.We shall prove that any projective limit of MV-spaces in the category of spectral spaces is an MV-space. In this way, we obtain new classes of MV-spaces related to some preservation properties of the Belluce functor.