Showing papers on "Banach space published in 1981"

On the Existence of Positive Solutions of Semilinear Elliptic Equations

Abstract: In this paper we study the existence of positive solutions of semilinear elliptic equations. Various possible behaviors of nonlinearity are considered, and in each case nearly optimal multiplicity results are obtained. The results are also interpreted in terms of bifurcation diagrams.

A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional

Abstract: We study Banach-space-valued martingale transforms and, in particular, characterize those Banach spaces for which the classical theorems of the real-valued case carry over. For example, if $B$ is a Banach space and $1 0$ and $\zeta(x, y) \leq | x + y | \text{if} | x | \leq 1 \leq | y |$.

Banach Spaces of Distributions of Wiener’s Type and Interpolation

Abstract: In the parallel paper [9] we have introduced “spaces of Wiener’s type”, a family of Banach spaces of (classes of) measurable functions, measures or distributions on locally compact groups. The elements of these spaces are characterized by — what we call — the global behaviour of certain of their local properties. In the present paper it is to be shown that interpolation methods can be applied to these spaces in a very natural way. Using the results on interpolation it is not difficult to extend various theorems of analysis to the setting of Wiener-type spaces. As illustration we present a version of the Hausdorff — Young inequality for locally compact abelian groups. As a consequence, one obtains a sharpened version of Soboley’s embedding theorem.

Espaces de Banach stables

Abstract: We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofLp(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containslp, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL1 containslp, for somep in the interval [1, 2].

Inequalities for b-valued random vectors with applications to the strong law of large numbers

Abstract: Analogues of the Marcinkiewicz-Zygmund and Rosenthal inequalities for Banach space valued random vectors are proved. As an application some results on the strong law of large numbers are obtained. It is proved that the Marcinkiewicz SLLN holds for every $p$-integrable, mean zero $B$-valued $\mathrm{rv}$ if and only if $B$ is of Rademacher type $p(1 \leq p < 2)$.

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

Abstract: We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS n =(I+T+…+T n−1 )/n, then lim n ‖S n (x)−TS n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping.

A non-reflexive Grothendieck space that does not containl ∞

Abstract: A compact spaceS is constructed such that, in the dual Banach spaceC(S)*, every weak* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol ∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl ∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.

Some Results on the LIL in Banach Space with Applications to Weighted Empirical Processes

Abstract: We examine the cluster set of $S_n/a_n$ for Banach space valued random variables, and investigate the relationship between the central limit theorem and the law of the iterated logarithm in this setting. In the case of Hilbert space valued random variables, necessary and sufficient conditions are given for the law of the iterated logarithm. Some interesting examples are also included. We then apply our results to weighted empiricals both in the supremum norm and the $L^2\lbrack 0, 1\rbrack$ norm.

Geometry and probability in Banach spaces

Abstract: Type and cotype for a Banach space p-summing maps.- Pietsch factorization theorem.- Completely summing maps. Hilbert-Schmidt and nuclear maps.- p-integral maps.- Completely summing maps: Six equivalent properties. p-Radonifying maps.- Radonification Theorem.- p-Gauss laws.- Proof of the Pietsch conjecture.- p-Pietsch spaces. Application: Brownian motion.- More on cylindrical measures and stochastic processes.- Kahane inequality. The case of Lp. Z-type.- Kahane contraction principle. p-Gauss type the Gauss type interval is open.- q-factorization, Maurey's theorem Grothendieck factorization theorem.- Equivalent properties, summing vs. factorization.- Non-existence of (2+?)-Pietsch spaces, Ultrapowers.- The Pietsch interval. The weakest non-trivial superproperty. Cotypes, Rademacher vs. Gauss.- Gauss-summing maps. Completion of grothendieck factorization theorem. TLC and ILL.- Super-reflexive spaces. Modulus of convexity, q-convexity "trees" and Kelly-Chatteryji Theorem Enflo theorem. Modulus of smoothness, p-smoothness. Properties equivalent to super-reflexivity.- Martingale type and cotype. Results of Pisier. Twelve properties equivalent to super-reflexivity. Type for subspaces of Lp (Rosenthal Theorem).

A characterization of complete metric spaces

Abstract: A general formulation of the completeness argument used in the Bishop-Phelps Theorem and many other places has been given by Ekeland. It is shown that Ekeland's formulation characterizes complete metric spaces. A central idea in the proof of the Bishop-Phelps Theorem is the use of norm completeness and a partial ordering to produce a point where a linear functional attains its supremum on a closed bounded convex set. In fact, this completeness technique is useful in many situations as has been described in the surveys of Phelps [7] and Brezis-Browder [1]. Recently Ekeland [3] has given a very general formulation of this technique and has applied it to a wide variety of problems [4]. In the present note we show that Ekeland's formulation is actually equivalent to completeness for metric spaces. Ekeland's Theorem may be stated as follows: THEOREM 1. Let (M, d) be a complete metric space, and F: M -R U { + oo} a lower semicontinuous function, F 5 + x, bounded from below. Let > 0 be given and a point u & M such that

Entropy numbers of diagonal operators with an application to eigenvalue problems

Abstract: One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five "simple" examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These discoveries showed that a separable infinite-dimensional Banach space is isomorphic to a complemented subspace of LP if and only if it is isomorphic to 12 or is an "EP-space", that is, equal to the closure of an increasing union of finite-dimensional spaces uniformly close to 1'P's. By making crucial use of statistical independence, the second author produced several more examples in [19], and the third author built infinitely many non-isomorphic examples in [23]. These discoveries left unanswered: Does there exist a Xp and infinitely many non-isomorphic Xp-complemented subspaces of LP (equivalently, are there infinitely many separable Ep A-spaces for some X depending on p)? We answer these questions by obtaining uncountably many non-isomorphic complemented subspaces of LP.* Before our work, it was suspected that every EP-space nonisomorphic to LP embedded in (12e 12e ... )P (for 2 < p < oc) (see Problem 1 of [23]). Indeed, all the known examples had this property. However our results show that there is no universal fp-space besides LP. To obtain these results, we use rather deep properties of martingales together with a new ordinal index, called the local LP-index, which assigns "large" countable ordinals to any

An ordinal $L^p$-index for Banach spaces, with application to complemented subspaces of $L^p$

Abstract: One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five "simple" examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These discoveries showed that a separable infinite-dimensional Banach space is isomorphic to a complemented subspace of LP if and only if it is isomorphic to 12 or is an "EP-space", that is, equal to the closure of an increasing union of finite-dimensional spaces uniformly close to 1'P's. By making crucial use of statistical independence, the second author produced several more examples in [19], and the third author built infinitely many non-isomorphic examples in [23]. These discoveries left unanswered: Does there exist a Xp and infinitely many non-isomorphic Xp-complemented subspaces of LP (equivalently, are there infinitely many separable Ep A-spaces for some X depending on p)? We answer these questions by obtaining uncountably many non-isomorphic complemented subspaces of LP.* Before our work, it was suspected that every EP-space nonisomorphic to LP embedded in (12e 12e ... )P (for 2 < p < oc) (see Problem 1 of [23]). Indeed, all the known examples had this property. However our results show that there is no universal fp-space besides LP. To obtain these results, we use rather deep properties of martingales together with a new ordinal index, called the local LP-index, which assigns "large" countable ordinals to any

More on convergence in unitary matrix spaces

Abstract: Let E be a symmetric sequence space satisfying the Radon-Riesz Property { lix11-+ lxii and x. x weakly) X llx-xll 0, then the same is true for the associated unitary matrix space CE. Let E be a symmetric sequence space, i.e., a Banach space of sequences so that the standard unit vectors {e,,})=I (defined by e,(j) = 8nj) form a 1-symmetric, normalized basis of E. The unitary matrix space CE associated with E is the Banach space of all compact operators x on 12 for which s(x) E E, normed by IIXIICE = IIS(x)IIE Here s(x) = (sn(x)) is the sequence of s-numbers of x, i.e., the eigenvalues of (x*x)'/2, arranged in a nonincreasing ordering, counting multiplicity. In the recent paper [1] we reduce the study of certain properties of CE to the study of the analogous properties of E. One consequence is the following result which characterizes convergence in CE in terms of convergence in E (see [2]): If {x") and x are elements of CE, then IIxn XIICE ->0 if and only if IIS(Xn) S(x)IIE O and xn -x weakly. In this note we apply [1] to study some related convergence property, and show that it extends from E to CE. DEFINITION. A Banach space X is said to have the Radon-Riesz Property (RRP, in short) if IIx,AI -X lxii and xn -* x weakly imply lIxn xll -> Ofor all {xj) and x in X. THEOREM I. Let E be a symmetric sequence space. Then E has the RRP if and only if CE has the RRP. This theorem extends the known result for the case E = 1p, I < p < oo, (see [3]; or apply the uniform convexity of the spaces C. = Ci,, [4]), and answers affirmatively a question of B. Simon. If CE has the RRP, then the same is true for E which is isometric to the subspace of CE consisting of all diagonal matrices. So the point in the theorem is that the RRP extends from E to CE. Received by the editors June 4, 1980 and, in revised form, July 18, 1980. AMS (MOS) subject classifications (1970). Primary 47D15; Secondary 46B99, 47B10.

Subspaces of ¹, via random measures

Abstract: It is shown that every subspace of LI contains a subspace isomorphic to some lq. The proof depends on a fixed point theorem for random measures.

A lattice renorming theorem and applications to vector-valued processes

Abstract: A norm, 11 11, on a Banach space E is said to be locally uniformly convex if IIxnII -lxii and IIx, + xlI -* 211xII implies that xn -* x in norm. It is shown that a Banach lattice has an (order) equivalent locally uniformly convex norm if and only if the lattice is order continuous. This result is used to reduce convergence theorems for (lattice-valued) positive martingales and submartingales to the scalar case. 0. Introduction. A norm, j, on a Banach space E is said to have the Kadec-Klee property (sometimes property (H)) if whenever xn -* x weakly and llx,ll -* lIx ll, then x0 -* x strongly. The aim of this paper is to show that if E has an equivalent Kadec-Klee norm then one may obtain convergence theorems for E-valued random processes (XJ) whenever j IXjj is a well-known real-valued convergent process and whenever the limit can be identified in the Banach space. For instance, the classical Kadec renorming theorem for separable Banach spaces gives the convergence of vector-valued martingales, uniform amarts and additive processes (ergodic theorem) since their norms are real-valued submartingales, amarts and subadditive processes respectively while the identification of the limit requires the RadonNikodym property on the space for the first two processes, the limit of the third process exists in any Banach space. If now, we consider positive submartingales and subadditive processes valued in a Banach lattice, one needs that the equivalent Kadec-Klee norm be also a lattice norm in order to conclude that the norms of these processes are real-valued submartingales and subadditive processes respectively. ?1 deals with the existence of such a lattice renorming while in ?1I we show how this leads to a unified approach for proving the almost sure convergence of the processes mentioned above. I. Renorming order continuous lattices. A norm, jj 1, on a Banach space E is said to be locally uniformly convex if jjx,jj --I jjxII and IIx + xnjj -2j1xII imply that xn-* x strongly. This notion is clearly stronger than the Kadec-Klee property. Received by the editors October 15, 1979 and, in revised form, February 11, 1980; presented to the Society, San Antonio, January 1980. AMS (MOS) subject classifications (1970). Primary 46B99, 60G99.

Isometries between Banach spaces of Lipschitz functions

Abstract: The Banach spaces Lip a (S, Δ), lip a (S, Δ), Lip a (S, Δ;s 0) and lip a (S, Δ;s 0) of Lipschitz functions are defined. We shall identify the extreme points of the unit balls in their corresponding dual spaces and make use of them to present a complete characterization of the isometries between these function spaces.

Morse theory by perturbation methods with applications to harmonic maps

Abstract: There are many interesting variational problems for which the PalaisSmale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential. The abstract Morse theory of Palais and Smale [9] can be applied with success to many problems in the calculus of variations to get existence theorems for stationary or critical points of integrals from topological information. However, there is a natural limitation in these applications in that the conditions are stated in terms of a fixed function space (which will usually be a Sobolev space). The integral must both be differentiable and satisfy a norm convergence property, which Palais and Smale call condition C, in the same space of functions. In this article we introduce a perturbation method which circumvents this difficulty in some variational problems. This technique may be applied in particular to harmonic mappings, and we include this somewhat technical discussion in the last section. The author is grateful to G. K. Francis and R. S. Palais for a number of helpful conversations. 1. Palais-Smale Morse theory. In this article, B will denote a Banach manifold with at least a C2 structure, modeled on a Banach space with at least C2 partitions of unity. We assume that this manifold is equipped with a C1 Finsler structure in which the manifold is complete. If a function f is defined and differentiable on B, ldfxI = maXoVET(B) Idf (v)I Iv2 is well defined. We say that the function f satisfies Palais-Smale condition C, if for any set S c B on which If(S)I is bounded and Idf(S)I is not bounded away from zero, the closure S contains a critical point of f (a point x where dfx = 0). We have the following theorem due to Palais and Schwartz [10], [12]. (1.1) THEOREM. Let B be a complete C2 Finsler manifold (without boundary) andf: B --*R a C2 function satisfying Palais-Smale condition C. Then f satisfies a Lusternik-Schnirelman theory. In particular (a) If f is bounded below it takes on its minimum on every component of B and there are at least as many critical points as cat(B). Received by the editors March 13, 1979 and, in revised form, September 26, 1979. 1980 Mathematics Subject Classification Primary 49F15; Secondary 58E05. 'Research supported in part by the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0462/$04.75 569 This content downloaded from 157.55.39.159 on Sun, 18 Sep 2016 06:10:27 UTC All use subject to http://about.jstor.org/terms

Equivalence of Nonlinear Complementarity Problems and Least Element Problems in Banach Lattices

Abstract: A class of not necessarily linear operators A: V → V* is introduced, where the Banach space V and its dual V* carry dual vector-lattice orderings ≥. These operators, called Z-maps, generalize the n × n real matrices with nonnegative off-diagonal elements. If A is a strictly monotone Z-map with certain regularity and growth conditions, and if F denotes the set of all vectors v ∈ V for which v ≥ 0, and Av ≥ 0, then it is shown that the complementarity problem, to find v ∈ F such that = 0, and the least element problem, to find v ∈ F with v ≤ w for all w ∈ F, have the same unique solution. Some other problems equivalent to these, and some examples are discussed.

On the structure of non-weakly compact operators on Banach lattices

Abstract: In this paper we study extensions of theorems of Hagler and Stegall on LP~ spaces [11] and of Rosenthal [24, 25], and Pelczynski [21] on C(K) spaces to more general Banach spaces with some lattice structure which do not contain complemented copies of I r We show that i fX is a separable Banach space such that Co does not embed into X* and T is a bounded linear operator from X into some Banach space Y such that T* Y* is not separable then (i) if X has local unconditional structure h la Gordon and Lewis [7].