Showing papers on "Banach space published in 1979"

Trace ideals and their applications

Abstract: Preliminaries Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for $\mathcal J_P$ Trace, determinant, and Lidskii's theorem $f(x)g(-i abla)$ Fredholm theory Scattering with a trace condition Bound state problems Lots of inequalities Regularized determinants and renormalization in quantum field theory An introduction to the theory on a Banach space Borel transforms, the Krein spectral shift, and all that Spectral theory of rank one perturbations Localization in the Anderson model following Aizenman-Molchanov The Xi function Addenda Bibliography Index.

Topological Vector Spaces II

Abstract: § 14 contains the elementary theory of normed spaces and Banach spaces. A number of classical examples are discussed, to which we shall refer time and again in the later parts of the book.

Regularity and stability for the mathematical programming problem in Banach spaces

Abstract: This paper deals with a regularity assumption for the mathematical programming problem in Banach spaces. The attractive feature of our constraint qualification is the fact that it can be considered as a condition on the active part only of the constraint, and that it is preserved under small perturbations. Moreover, we show that our condition is “almost” equivalent to the existence of a non-empty and weakly compact set of Lagrange multipliers. The main step in the proof of our results is a generalization of the open mapping theorem.

Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces

Abstract: This study is devoted to constrained optimization problems in Banach spaces. We present different properties of tangent cones associated with an arbitrary subset of a Banach space and establish correlations with some of the existing results. In absence of both differentiability and convexity assumptions on the functions involved in the optimization problem, the consideration of these tangent cones and their polars leads us to introduce new concepts in nondifferentiable programming. Necessary optimality conditions are first developed in a general abstract form; then these conditions are made more precise in the presence of equality constraints by introducing the concept of normal subcone.

A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces

Abstract: The following theorem is proven:if E is a uniformly rotund Banach space with a Frechet differentiable norm, C is a bounded nonempty closed convex subset of E, and T: C→C is a contraction, then the iterates {Tnx} are weakly almost-convergent to a fixed-point of T.

M-Structure and the Banach-Stone Theorem

Abstract: Preliminaries.- L-projections and M-projections.- M-Ideals.- The centralizer.- Function modules.- M-Structure of some classes of Banach spaces.- Remarks.- The Banach-Stone theorem.- The Banach-Stone property and the strong Banach-Stone property.- Centralizer-norming systems.- M-structure of Co(M,X).- Generalizations of the Banach-Stone theorem.- Remarks.

Twisted sums of sequence spaces and the three space problem

Abstract: In this paper we study the following problem: given a complete locally bounded sequence space Y, construct a locally bounded space Z with a subspace X such that both X and Z/X are isomorphic to Y, and such that X is uncomplemented in Z. We give a method for constructing Z under quite general conditions on Y, and we investigate some of the properties of Z. In particular, when Y is lp (1

Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces

Abstract: LetA be anm-accretive operator in a Banach spaceE. Suppose thatA −10 is not empty and that bothE andE * are uniformly convex. We study a general condition onA that guarantees the strong convergence of the semigroup generated by—A and of related implicit and explicit iterative schemes to a zero ofA. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also establish strong convergence under the assumption that the interior ofA −10 is not empty. In Hilbert space this result is due to H. Brezis.

On rational approximations of semigroups

Abstract: We show that if $r^n (hA),nh = t$, is an A-acceptable rational approximation of a strongly continuous semigroup $e^{tA} $ on a Banach space, then for t bounded, $\| {r^n (hA)} \| \leqq Cn^{{1/2}} $, with certain improvements under additional hypotheses on r. We also discuss the convergence of $r^n (hA)v$ to $e^{tA} v$ as $h \to 0$ under various assumptions on r and v.

Constructive techniques for accretive and monotone operators

Abstract: Publisher Summary This chapter describes the constructive techniques for accretive and monotone operators. It highlights new and recent convergence results, error estimates, applications, and open problems. The chapter presents a new result for strongly accretive operators. The chapter presents an assumption that E is a uniformly convex Banach space with a uniformly convex dual and a duality mapping that is weakly sequentially continuous at zero and A ⊂ E × E is an w-accretive operator with a zero. Zeros of certain monotone operators correspond to solutions of certain partial differential equations. Convergent schemes may be used to prove existence. In probabilistic analysis, they may be used to establish existence of random solutions to random equations.

Projections onto Hilbertian subspaces of Banach spaces

Abstract: In this paper we obtain new estimates for the relative projection constants of subspaces of a Banach spaceY in terms of geometrical properties ofY. Our method gives thatK-convex spaces are locally π-Euclidean. We also get a version of Maurey’s extension theorem for spaces of typep<2.

Existence of fixed points of nonexpansive mappings in certain Banach lattices

Abstract: A fixed point theorem for nonexpansive mappings in dual Banach spaces is proved. Applications in certain Banach lattices are given. 1. Suppose K is a subset of a Banach space X and T: K -* K is a nonexpansive mapping, i.e. II T(x) T(y)jj S lix -yIj, x, y E K. A wellknown theorem due to Kirk [1] states that, if K is convex weakly compact (weak* compact when X is a dual space) and has normal structure, then T has a fixed point in K. In particular, if X = LP (1 0. Here and in the sequel V and A denote the least upper bound and the greatest lower bound respectively. X is said to be order complete if each set A C X with an upper bound has a least upper bound. A complex AM-space is defined as the complexification of an AM-space. Suppose X is an order complete AM-space with unit (i.e. an element e such that the unit ball at zero is the order interval [e, e]); then X is isometrically lattice isomorphic to the space CR (S) of all continuous real-valued functions defined on a compact Stonian space S. For these and other facts about Banach lattices we refer to Schaefer's book [4]. Received by the editors March 19, 1978. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 46B99, 46E05.

Fixed point iterations using infinite matrices

Abstract: This chapter discusses fixed point iterations using infinite matrices. It presents an assumption as per which there is a normed linear space X ; C is a nonempty, closed, bounded and convex subset of X ; T : C → C is a mapping with at least one fixed point; and A is an infinite matrix. Given the iteration scheme x 0 = x 0 in C ; x n +1 = Tx n , n = 0, 1, 2 …, the chapter discusses the restriction on the matrix A that is necessary and/or sufficient to guarantee that the iteration scheme converges to a fixed point of T . Using these iteration schemes, results have been obtained for certain class of infinite matrices. This chapter presents the generalizations of several of these results. It presents a proof of a theorem for the solution of operator equations in a Banach space involving generalized contraction mappings and also presents a few results as corollaries.

Generic differentiability of Lipschitzian functions

Abstract: It is shown that, in separable topological vector spaces which are Baire spaces, the usual properties that have been introduced to study the local "first order" behaviour of real-valued functions which satisfy a Lipschitz type condition are "generically" equivalent and thus lead to a unique class of "generically smooth" functions. These functions are characterized in terms of tangent cones and directional derivatives and their "generic" differentiability properties are studied. The results extend some of the well-known differentiability properties of continuous convex functions. Introduction. This paper is devoted to the investigation of generic smoothness properties for locally Lipschitzian real-valued functions, the domain of which is an open subset of a topological vector space. There has been a recent revival of this problem in the restricted case of convex functions [2], [3], [9], [10], [14]; the first result along this vein seems to be due to E. Asplund and asserts that every continuous convex function defined on an open convex subset of a Banach space is Gateaux (or even Frechet) differentiable on a dense G8 subset of its domain, provided the existence of a dual norm which enjoys some nice geometrical properties. Unfortunately, well-known examples show that such a property no longer holds for an arbitrary locally Lipschitzian real-valued function. Therefore, many authors have sought a generalization of the classical Rademacher theorem involving a new concept of theoric "measure-zero" set [1], [6], [13]. However, a lot of work can be done to extend the generic differentiability properties to a large class of real-valued functions. The main contribution of this note is to give, at least in "small" spaces (here it means separable spaces), a characterization of these funcions, expressed both in terms of directional derivatives and tangent cones. The first section deals with the notion of a generalized gradient introduced by F. H. Clarke, [7], [8], from whom I have benefitted greatly. Theorem (1.7) is the cornerstone to our development and allows us to dispense with convexity assumptions in the proofs of Theorems (2.1) and (2.2). In the next section, we discuss the relation between smoothness properties of locally Received by the editors April 27, 1978. AMS (MOS) subject classifications (1970). Primary 58C20; Secondary 26A24, 26A27. ( 1979 American Mathematical Society 0002-9947/79/0000-0555 /$06.00 125 This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:22:12 UTC All use subject to http://about.jstor.org/terms

Unconditional bases and martingales in LP(F)

Abstract: Here we describe our results and their background: terminology (mostly standard) is denned in Section 2. Throughout, F is a separable Banach space, 1 ≤ p < ∞ and Lp(F) is the space of measurable functions [0,1] → F with P-integrable norms. Given a ‘nice’ property P for Banach spaces, we may formulate the conjecture: Lp(F) satisfies P if and only if both F and Lp (= LP(ℝ)) satisfy P. This conjecture is known to be true for various specific properties, for example the Radon–Nikodym property ((4), section 5·4); reflexivity ((4), corollary 4·1·2); super-refiexivity ((12), proposition 1·2); B-convexity ((14), p. 200); and the properties of not containing copies of c0 (6) and l1 (13). The object of this paper is to demonstrate that the conjecture is false for the property of having an unconditional basis – this answers a question in (4).

A Generalization of Uniformly Rotund Banach Spaces

Abstract: Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ X Thus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real L p spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for The space is said to be uniformly rotund if for each 0 we have .

Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces

Abstract: The following theorem is proved: Let S ( t ), t ≧0 be a dynamical system in an infinite dimensional Banach space X such that S ( t ) = S 1 ( t )+ S 2 ( t ) for t ≧0, where (1) uniformly in bounded sets of x in X , and (2) S 2 ( t ) is compact for t sufficiently large. Then, if the orbit { S ( t ) x : t ≧0} of x ∈ X is bounded in X , it is precompact in X . Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

Examples for the nonlocally convex three space problem

Abstract: A simple way to obtain certain examples of locally bounded spaces E of the following kind is described: E is nonlocally convex but contains a locally convex subspace K such that E/K is locally convex. The purpose of this note is to give a quite simple construction which proves the following: THEOREM 1, (Independently proved by Kalton [5]). There is a separated locally bounded space E with an uncomplemented one-dimensional subspace L such that E/L is isomorphic to 11. THEOREM 2. Indeed, there is an uncountable class of such spaces E of which no one can be mapped into another by a continuous linear mapping which does not annihilate the uncomplemented line. THEOREM 3A. There is a locally bounded space E with a subspace K such that K is isomorphic to 1' and E/K to 11, and such that for any infinite-dimensional subspace E1 c E \ K, the subspace E1 + K fails to be locally convex. THEOREM 3B. Let A be a set of the cardinality of the continuum. There is a locally bounded space E with a subspace K such that K is isomorphic to 1 '(A) and E/K to 11, and such that for any infinite-dimensional subspace El c E \ K, the subspace El + K fails to be separated by its dual. Kalton [5] also proves that in Theorem 1, 1P cannot be replaced by 1P for any p #& 1 (O < p < ox). For some further recent related results, see Dierolf [1], Kalton and Peck [6], and Roberts [9]; cf. also Ribe [7]. We are thus dealing with the so-called three space problem, which is of current interest also in the pure Banach space setting; some results have been given by Enflo, Lindenstrauss, and Pisier [2]. In the desire to grasp the nature of this problem it is apparently well motivated to study cases where transparent solutions are possible. Received by the editors November 9, 1977 and, in revised form, April 14, 1978. AMS (MOS) subject classifications (1970). Primary 46A10; Secondary 46B05.

Stable measures and central limit theorems in spaces of stable type

Abstract: Let X be a symmetric random variable with values in a quasinormed linear space E. X satisfies the central limit theorem on E with index p, 0 < p < 2, if E (n-I/P(Xi + * + X")) converges weakly to some probability measure on E. Hoffman-Jdrgensen and Pisier have shown that Banach spaces of stable type 2 provide a natural environment for the central limit theorem with index p = 2. In this paper we show that, for 0 < p < 2, quasi-normed linear spaces of stable type p provide a natural environment for the central limit theorem with indexp. A similar result holds also for the weak law of large numbers with index p. 0. Introduction. Let X be a symmetric random variable with values in a quasi-normed linear space E. We say that X satisfies the central limit theorem on E with index p, 0