Showing papers on "Banach space published in 1985"

Approximation Theory in Tensor Product Spaces

Abstract: An introduction to tensor product spaces.- Proximinality.- The alternating algorithm.- Central proximity maps.- The diliberto-straus algorithm in C(S x T).- The algorithm of von golitschek.- The L 1-version of the diliberto-straus algorithm.- Estimates of projection constants.- Minimal projections.- Appendix on the bochner integral.- Appendix on miscellaneous results in banach spaces.

Quadratic equations and applications to Chandrasekhar's and related equations

Abstract: A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced. The results are then applied to solve Chandrasekhar's integral equation and related equations without the usual positivity assumptions.

Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces

Abstract: © Annales de l’institut Fourier, 1985, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the Cauchy problem for the nonlinear Boltzmann equation global existence uniqueness and asymptotic stability

Abstract: The analysis of the initial value problem for the nonlinear Boltzmann equation is considered in this paper. A theorem defining global existence and uniqueness for initial data which decay at infinity with an inverse power law is the main result of this work and is obtained by suitable application of fixed point theorems in Banach spaces. The theorem also defines the asymptotic stability of the solutions.

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

Abstract: It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL 0(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl p (respectivelyL p (0, 1)), 2

Relations Between the $s$-Selfdecomposable and Selfdecomposable Measures

Abstract: The classes of the $s$-selfdecomposable and decomposable probability measures are related to the limit distributions of sequences of random variables deformed by some nonlinear or linear transformations respectively. Both are characterized in many different ways, among others as distributions of some random integrals. In particular we get that each selfdecomposable probability measure is $s$-selfdecomposable. This and other relations between these two classes seem to be rather unexpected.

Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure

Abstract: Dans un espace de Banach X, on etudie quand reflexivite et structure normale sont suffisantes pour assurer, pour de bon γ>1, la propriete de point fixe pour des auto-applications uniformement γ-lipschitziennes

Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence

Abstract: The aim of this note is to discuss the convergence properties of various methods for the approximate solution of ill-posed equations. If an operator T between Banach spaces has a non-closed range, then there exists no linear uniformly convergent approximation, method, but at most pointwise approximation methods. for the approximate solution of an equation (1) Tx = y. These methods in general converge arbitrarily slow, i.e. there may exist some y for which the approximations have a good convergence rate, but in general for each order of convergence there are right hand sides y with a worse convergence rate. This phenomenon is not restricted to ill-posed equations. Recently I have shown [121 that many common approximation schemes for integral equations of the second kind show arbitrarily slow convergence, too.

Convexification in limit laws of random sets in banach spaces

Abstract: A convexification argument is offered which implies that the strong law of large numbers for random compact sets in a Banach space is valid without a convexity assumption.

Uncountable constructions for B.A. e.c. groups and banach spaces

Abstract: This paper has two aims: to aid a non-logician to construct uncountable examples by reducing the problems to finitary problems, and also to present some construction solving open problems. We assume the diamond forℵ1 and solve problems in Boolean algebras, existentially closed groups and Banach spaces. In particular, we show that for a given countable e.c. groupM there is no uncountable group embeddable in everyG\(G L_{L,\omega } \)-equivalent toM; and that there is a non-separable Banach space with noℵ1 elements, no one being the closure of the convex hull of the others. Both had been well-known questions. We also deal generally with inevitable models (§4).

Mathematics of Kalman-Bucy filtering

Abstract: 1. Elements of Probability Theory.- 1.1 Probability and Probability Spaces.- 1.1.1 Measurable Spaces, Measurable Mappings and Measure Spaces.- 1.1.2 Probability Spaces.- 1.2 Random Variables and "Almost Sure" Properties.- 1.2.1 Mathematical Expectations.- 1.2.2 Probability Distribution and Density Functions.- 1.2.3 Characteristic Function.- 1.2.4 Examples.- 1.3 Random Vectors.- 1.3.1 Stochastic Independence.- 1.3.2 The Gaussian N Vector and Gaussian Manifolds.- 1.4 Stochastic Processes.- 1.4.1 The Hilbert Space L2(?).- 1.4.2 Second-Order Processes.- 1.4.3 The Gaussian Process.- 1.4.4 Brownian Motion, the Wiener-Levy Process and White Noise.- 2. Calculus in Mean Square.- 2.1 Convergence in Mean Square.- 2.2 Continuity in Mean Square.- 2.3 Differentiability in Mean Square.- 2.3.1 Supplementary Exercises.- 2.4 Integration in Mean Square.- 2.4.1 Some Elementary Properties.- 2.4.2 A Condition for Existence.- 2.4.3 A Strong Condition for Existence.- 2.4.4 A Weak Condition for Existence.- 2.4.5 Supplementary Exercises.- 2.5 Mean-Square Calculus of Random N Vectors.- 2.5.1 Conditions for Existence.- 2.6 The Wiener-Levy Process.- 2.6.1 The General Wiener-Levy N Vector.- 2.6.2 Supplementary Exercises.- 2.7 Mean-Square Calculus and Gaussian Distributions.- 2.8 Mean-Square Calculus and Sample Calculus.- 2.8.1 Supplementary Exercise.- 3. The Stochastic Dynamic System.- 3.1 System Description.- 3.2 Uniqueness and Existence of m.s. Solution to (3.3).- 3.2.1 The Banach Space L2N(?).- 3.2.2 Uniqueness.- 3.2.3 The Homogeneous System.- 3.2.4 The Inhomogeneous System.- 3.2.5 Supplementary Exercises.- 3.3 A Discussion of System Representation.- 4. The Kalman-Bucy Filter.- 4.1 Some Preliminaries.- 4.1.1 Supplementary Exercise.- 4.2 Some Aspects of L2 ([a, b]).- 4.2.1 Supplementary Exercise.- 4.3 Mean-Square Integrals Continued.- 4.4 Least-Squares Approximation in Euclidean Space.- 4.4.1 Supplementary Exercises.- 4.5 A Representation of Elements of H (Z, t).- 4.5.1 Supplementary Exercises.- 4.6 The Wiener-Hopf Equation.- 4.6.1 The Integral Equation (4.106).- 4.7 Kalman-Bucy Filter and the Riccati Equation.- 4.7.1 Recursion Formula and the Riccati Equation.- 4.7.2 Supplementary Exercise.- 5. A Theorem by Liptser and Shiryayev.- 5.1 Discussion on Observation Noise.- 5.2 A Theorem of Liptser and Shiryayev.- Appendix: Solutions to Selected Exercises.- References.

On linear Volterra equations in Banach spaces

Abstract: In this paper we present the basic theory for a class of Volterra differential-integral equations of convolution type in Banach spaces. We show that existence of resolvent operator for such an equation is equivalent to its wellposedness, and we obtain a Hille-Yosida theorem. Unfortunately, this result is not easy to apply and therefore it is important to have peturbation theorems available. We present a result of this type which also contains an existence theorem, and show by means of several examples that it cannot be improved.

Multilinear mappings of nuclear and integral type

Abstract: We define vector-valued multilinear mappings of nuclear and integral type and establish conditions for its coinciding spaces. Introduction. If E and F are Banach spaces then LN(E, F), Lpl(E, F), and LGJ(E, F) denote the Banach spaces of all nuclear, Pietsch-integral, and Grothendieck-integral linear mappings from E into F endowed with their respective norms. We always have the continuous inclusions LN(E, F) c Lpl(E, F) C LG(E, F), and we are interested in finding conditions on E or F which guarantee the opposite inclusions. Let us recall the following results from the book of Diestel and Uhl [3]. THEOREM A [3, Theorem VI. 4.8]. A Banach space F has the Radon-Nikodym property if and only if for every Banach space E, each T E Lpl(E, F) is nuclear. In this case the identity mapping, LN(E, F) -* Lpl(E, F) is an isometry. THEOREM B [3, Theorem VIII. 4.6]. Let E be a Banach space whose dual E * has the approximation property. Then E * has the Radon-Nikodym property if and only if for every Banach space F, each T E LG(E, F) is nuclear. In this case the identity mapping LN(E, F) LGIJ(E, F) is an isometry. Since the identity Lpl(E, F) = LG(E, F) holds, e.g. whenever F is a dual space, Theorem B may be regarded as a sort of dual of Theorem A. In this paper we establish the following variant of Theorem B, which does not involve the approximation property. THEOREM 1.3. The dual E * of a Banach space E has the Radon-Nikodym property if and only if, for every Banach space F, each T E Lpl(E, F) is nuclear. In this case the identity mapping LN(E, F) -* Lpl(E, F) is an isometry. Curiously enough, Theorem A does not extend to the case of bilinear mappings, as we shall see in Remark 2.4; but with the obvious notation, Theorem 1.3 can be extended to the case of bilinear mappings as follows: Received by the editors March 25, 1984. 1980 Mathemcatics Subject Classification. Primary 15A69; Secondary 46B22, 46G10. 'Research supported in part by FAPESP (Brazil) when the author was visiting Kent State University. Permatnent address: Universidade de S-ao Paulo, Instituto de Matematica e Estatistica, Caixa Postal 20570, 05508-Sao Paulo, Brasil ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 33 This content downloaded from 207.46.13.124 on Sun, 11 Sep 2016 05:06:12 UTC All use subject to http://about.jstor.org/terms

On the Role of the Baire Category Theorem and Dependent Choice in the Foundations of Logic

Abstract: The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Cech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The RasiowaSikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces. The relevance of the Baire Category Theorem to the fundamental metalogical principle of deductive completeness has long been known. Rasiowa and Sikorski [1950], in their Boolean-algebraic proof of Gddel's completeness theorem for firstorder logic, applied the Baire Category Theorem to the compact Hausdorff Stone space of a Boolean algebra to obtain their celebrated lemma about the existence of ultrafilters respecting countably many meets. Grzegorczyk, Mostowski, and RyllNardzewski [1961] later adapted this approach to obtain the completeness theorem for co-logic, by applying the Baire theorem to the complete metric space of co-models of an w-complete theory. A similar argument may also be developed for omittingtypes theorems. The aim of this paper is to isolate that part of the Rasiowa-Sikorski Lemma that does not depend on the Ultrafilter Theorem. A result about the existence of certain filters is obtained that is dubbed "Tarski's Lemma" since it is closely allied to Tarski's algebraic proof of the Rasiowa-Sikorski Lemma, as reported by Feferman [1952]. It will be shown that in set theory without choice, Tarski's Lemma is equivalent to each of (i) the Baire Category Theorem for Cech-complete spaces, (ii) the Baire Category Theorem for complete metric spaces, (iii) the Principle of Dependent Choice, (iv) the existence theorem for generic sets of forcing conditions, and (v) a proof-theoretic principle which abstracts the technique introduced by Henkin [1949] for proving completeness proofs. From this follows a proof that the Rasiowa-Sikorski Lemma is equivalent to the conjunction of the Ultrafilter Theorem with the Baire Category Theorem for compact Hausdorff spaces. Throughout this paper, assertions that one statement implies another, or that two statements are equivalent (imply each other) will mean that the implications Received November 1, 1983; revised February 23, 1984. ? 1985, Association for Symbolic Logic 0022-48 12/85/5002-0016/$02. 1 0

Folds and cusps in Banach spaces, with applications to nonlinear partial differential equations. I

Abstract: On donne deux caracterisations abstraites de l'application coupelle G:R 2 ×E→R 2 ×E, (t,λ,ν)→(t 3 −λt,λ,ν) ou E est un espace de Banach reel

Characterization of some classes of operators on spaces of vector-valued continuous functions

Abstract: Let $K$K be a compact Hausdorff space and $E$E, $F$F Banach spaces with $L(E,F)$L(E,F) the space of bounded linear operators from $E$E into $F$F. If $C(K,E)$C(K,E) is the space of all continuous functions from $K$K into $E$E equipped with the sup-norm, then every operator $T\in L(C(K,E),F)$T∈L(C(K,E),F) has a representing measure $m$m of bounded semivariation on the Borel sets of $K$K with values in $L(E,F'')$L(E,F′′) such that $TF=\int_Kf\,dm$TF=∫Kfdm. If $T$T is a weakly compact operator, then $m$m has values in $L(E,F)$L(E,F), $m(E)$m(E) is weakly compact for each Borel set $E$E, and the semivariation of $m$m is continuous at $\varphi$φ. It is known that the converse of this statement does not hold in general, but does hold under additional assumptions. In particular, the authors show that the converse holds if $K$K is a dispersed space. They also show that, in a certain sense, the assumption that $K$K is a dispersed space is necessary; that is, if the converse of the statement above holds for every pair of Banach spaces $E,F$E,F then $K$K must be a dispersed space. A similar result holds for the class of unconditionally converging, Dunford-Pettis or Dieudonne operators.

A general equivalence theorem in the theory of discretization methods

Abstract: The Lax-Richtmyer theorem is extended to work in the framework of Stetter's theory of discretizations. The new result applies to both initial and boundary value problems discretized by finite elements, finite differences, etc. Several examples are given, together with a comparison with other available equivalence theorems. The proof relies on a generalized Banach-Steinhaus theorem. 1. Introduction. In this paper we extend the classical Lax-Richtmyer equivalence theorem (6), so as to cover in a simple way not only initial value problems, but also boundary value problems, mixed problems, etc. Our theory relies on a generalized Banach-Steinhaus theorem (9) and works (essentially) in the framework of Stetter (13). This set-up employs restriction operators to compare the true and discretized solutions, as distinct to those theories which use prolongation operators. (One of the oldest prolongation theories is probably that of Aubin, summarized in (16).) Our main result is given in Section 2. Sections 3 and 4 are devoted to examples and counterexamples. The former are meant to show the scope of our result and include the Galerkin method for boundary value elliptic problems and semidiscrete and fully discrete schemes for initial value problems. The counterexamples prove that the present hypotheses cannot be dispensed with. In particular, we show that a method which is consistent and convergent for all data in a Banach space may be unstable. The final section contains a comparison with other available equivalence theorems.

Inequalities for the Schatten p -norm II

Abstract: This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

Banach spaces embedding into L 0

Abstract: Our main result in this paper is that a Banach spaceX embeds intoL 1 if and only ifl 1(X) embeds intoL 0; more generally if 1≦p<2,X embeds intoL p if and only ifl p(X) embeds intoL 0.