Parabolic spaces

In the present paper we define parabolic spaces with simple fundamental Lie groups and we construct models of them, establish a relation between parabolic and reductive spaces, describe the topological structure of parabolic spaces with classical Lie fundamental groups.The introduction and Sec. 1 were written by B. A. Rozenfel'd, Sec. 2 by M. P. Zamakhovskii, Sec. 3 by T. A. Timoshenko.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 26, pp. 125–160, 1988.     

Study of families of monotone continuous functions on Tychonoff spaces

The main object of study is the space of all monotone continuous functions CM(X) on a connected Tychonoff space X endowed with the topology of pointwise (CM _p (X)) or uniform (CM(X)) convergence. Technical questions concerning restriction and extension of monotone functions are considered in Sec. 2. Conditions for CM(X) to separate the points of X and for CM(X) to contain only constant functions are found in Sec. 3. In Sec. 4, the linear structure of CM(X) is studied and all linear subspaces of CM(X) for a certain class of spaces X are described. In Sec. 5, conditions under which CM(X) is closed and nowhere dense in C _p (X) and C(X) are determined. The metrizability of CM _p (X) is considered in Sec. 6; necessary and sufficient metrizability conditions for various classes of spaces X are obtained. In Sec. 7, criteria for σ-compactness and the Hurewicz property in the class of spaces CM _p (X) are given. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.     

Parseval's equality in the abstract interpolation problem and the coupling of open systems. II

The constructions described in Sec. 1 are applied to the investigation of the abstract interpolation problem. The general solution of the problem is the characteristic function of an operator colligation, obtained by the closure of fixed colligation by means of an arbitrary colligation with definite exterior spaces. The complete integral representation of a nonnegative quadratic form is obtained by applying Parseval's equality, considered in Sec. 1.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 98–103, 1988.     

Inertia and eigenvalue relations between symmetrized and symmetrizing matrices for the real and the general field case

Starting from a theorem of Frobenius that every n×n matrix is the product of two symmetric ones, we study relations between the similarity invariants of a square matrix and the congruence invariants of its symmetric factors. Section 1 treats the real case, Sec. 2 the arbitrary field case, and Sec. 3 the indefinite inner product case for Krein spaces. The proofs are obtained from the real canonical pair form in Secs. 1 and 3 and from the recently found rational canonical pair form in Sec. 2, each time via combinatorial type arguments on weighted partitions of n. The resulting theorems typically give bounds for the elementary divisor structure of A in terms of the index or signature of one or both of its symmetric factors (or vice versa). Our results greatly extend and generalize the classic results of Klein, Loewy, Taussky, et al., and in Sec. 2 put new light on Waterhouse's recent characterization of hereditarily euclidean fields. A short survey on the history of the subject from the early 1800s on completes the paper.     

Theory of compositions

A survey of papers on the theory of compositions and their application to the investigation of manifolds in spaces with fundamental groups is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 10, pp. 117–145, 1978.I thank Érnst Gergardovich Neifel'd, whose paper [12] stimulated much of this survey, and who, moreover, showed me inestimable help in the formulation of my paper. In particular, he wrote Sec. 17.     

Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Multi-dimensional volumes and moduli of convexity in banach spaces

Summary Let E be a Banach space. Using the definition for the k-dimensional volume enclosed by k + 1 vectors due to Silverman [16], one can define the modulus of k-rotundity of E. In [22] it was shown that k-uniformly rotund Banach spaces are isomorphic to uniformly rotund spaces and, indeed, have some of the same isometric properties with respect to non-expansive and nearest-point maps. The present paper examines the modulus of k-rotundity more thoroughly. Included are a result on the asymptotic behavior of the moduli for l²; a generalization of Dixmier's Theorem on higher-duals of non-reflexive spaces; and an inequality relating the moduli of E^**/E and those of E. The modulus of2-rotundity is shown to be equivalent to one of the moduli defined by V. D. Milman [13] and a necessary and sufficient condition for an l^p-product of spaces to be 2-uniformly rotund is given.Some of the results of this paper are contained in the Ph. D. dissertation of the first author, written under the direction of the second author.     

Imbedding theorems of Besov spaces in Banach lattices

The paper examines imbeddings of Besov spaces B _{E, θ} ^ω in ideal spaces (Banach lattices) given that ω ∈ S_k). In particular, the symmetric hull of the space B _{E, θ} ^ω is described (E is a symmetric space), an inequality of different metrics is obtained, and imbeddings in Orlicz and Lorentz spaces and in some weighted spaces are studied. Most of the results are easily extended to the anisotropic case. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 69–82, 1987.     

K(o)the-Bochner空间的凸性

本文利用K(o)the函数空间的性质以及K(o)the函数空间与K(o)the-Bochner空间的关系,讨论了K(o)the-Bochner空间E(X)的凸性,主要结果如下:(a)给出E(X)的端点的充分条件,得到了E(X)严格凸的判据,相应地推广了Lp(μ,X)以及LΦ(X)的结果;(b)讨论了E(X)的弱局部一致凸和局部完全k-凸;(c)刻画了E(X)的强凸,给出了E(X)强凸的充要条件.     

赋序列范数的矢值Banach序列空间ss(E)的凸性

本文对赋序列范数的矢值Banach序列空间ss(E)的一些凸性进行了讨论,得到的主要结果如下: 1.ss(E)是局部一致凸的当且仅当ss和E是局部一致凸的; 2.ss(E)是强凸的当且仅当ss和E是强凸的; 3.设ss和ss*具有AK性质,则ss(E)是非常凸的当且仅当ss和E是非常凸的.     

An infinite dimensional version of the Paley-Wiener-Schwartz isomorphism

In the infinite dimensional case it is known from a Dineen-Nachbin counterexample that the natural generalization of the Paley-Wiener-Schwartz theorem is no longer valid: The image through the Fourier transform F of the space ?′(E) is made of entire functions on E′ that, besides the usual inequality, satisfy a more technical condition. We define and study here a dense subspace, denoted by E(E), of ?(E) with a proper topology such that F (E′(E)) is made of the entire functions on E′ that satisfy only the classical inequality. For this reason this space E(E) is suited for the study of convolution and partial differential equations in spaces of C^∞ functions on locally convex spaces.     

Post hulls of spaces of signatures

We argue that Post hulls of spaces of signatures are a natural generalization of Boolean hulls of spaces of orderings.     

On Isomorphisms of Relations Embedded into Each Other

In this paper, we consider analogs of the Cantor--Bernstein theorem for sets with binary relations. In Sec. 1, we prove an analog of this theorem for arbitrary binary relations; in Sec. 2, we consider an application; in Sec. 3, we study a class of relations with the Cantor-- Bernstein property and a class of exact relations, and prove that these classes are closed under certain operations.     

Measures on spaces of operators and isometries

One considers a question that arises at the investigation of isometric operators in vector-valued L^p -spaces. Let E, F be Banach spaces, let p>0, let μ be a probability Borel measure on the space on continuous linear operators from E into F such that for any e ε E one has $$\left\| e \right\|^p = \int {\left\| {Te} \right\|^p } d\mu \left( T \right)$$ . In the cases when: 1) E=C(K), K is a metric compactum, F is an arbitrary space, p>1 and 2) 2)E=F=L^q,p>1, q>1 q ?[p,2] it is Proved that the support of the measure μ is contained in the set of the operators that are scalar multiples of isometries. For E=C(K) one obtains an isomorphic analogue of this result: if the Banach-Mazur distance between C(K) and the P -sum of Banach spaces is small, then the distance between C(K) and one of the spaces is small.     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

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Über die Gleichstetigkeit von Mengen von linearen Abbildungen und eine Klasse von topologischen linearen Räumen

In the first part of this paper we proof the following theorem: Let E and F be topological linear spaces, α an infinite cardinal number, and H a set of linear mappings from E into F such that every subset G of H with cardinality |G|≤α is equicontinuous. Then H is equicontinuous on every linear subspace of E which is the closed linear hull of a family (B_L;L∈I), |I|≤α, of precompact subsets of E. In the second part we introduce the class of all topological linear spaces E with the following property: A set H of linear mappings from E into a topological linear space is equicontinuous, if every countable subset of H is equicontinuous. We show that this class is closed with respect to forming topological products and linear final topologies.     

Sobolev,Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval

Summary We consider various fractional properties of regularity for vector valued functions defined on an interval I. In other words we study the functions in the Sobolev spaces W^s,p(I;E), in the Nikolskii spaces N^s,p(I;E), or in the Besov spaces B _{s, p} (I; E). Theses spaces are defined by integration and translation, and E is a Banach space. In particular we study the dependence on the parameters s, p and , that is imbeddings for different parameters. Moreover we compare each space to the others, and we give Lipschits continuity, existence of traces and q-integrability properties. These results rely only on integration techniques.     

A Theorem on Multiplicators in Nonhomogeneous Hölder Spaces and Some of Its Applications

In Sec. 2, sufficient conditions of the boundedness of convolution operators with kernel m in a nonhomogeneous Hölder space are given in terms of the Fourier-image of m. After that, the results of Sec. 2 are used to prove the solvability in a Hölder space of the Cauchy problem for linear systems of the hydrodynamic type. Bibliography: 6 titles.     

Cohomologies of spaces of Schwartz test functions

We calculate cohomologies of the Schwartz algebras D and S, which are basic spaces (test-function spaces) of the distribution theory. In the process, we find special cohomologies of the quotient algebras E/D and M/S (E andMare also test-function spaces), which are quite unusual from the standpoint of the standard functional analysis and are interesting for theoretical and mathematical physics.     

Basic sequences in some regular Köthe spaces

In this paper (closed, linear) subspaces of nuclear Köthe spaces are investigated. This has been the topic of various papers by E. Dubinsky, D. Vogt, M. Alpseymen etc. We give complete characterizations for subspaces with basis of unstable Köthe spaces of type D₁, in particular unstable L_f (Drafilev) spaces of type d₁, and regular subspaces of L_f spaces of infinite type without any assumptions on the defining exponent sequence. The method used depends on applying a stability theorem on embeddings of L. Schwartz [17] and the Hall-Koenig theorem [10] on selection of distinct representatives as well as some construction methods of E. Dubinsky [9].