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1.
Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,
where π X is the canonical projection fromX *** ontoX *. We show some examples of Banach spaces not containingc 0, having the point of continuity property and satisfying the above inequality forr not necessarily equal to one. On the other hand, we prove that a Banach spaceX satisfying the above inequality fors=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition,X satisfies
wheneveru *,v *X * with ‖u *‖≤‖v *‖ and (x α * ) is a bounded weak* null net inX *, thenX can be renormed to satisfy the,M(r, 1) and theM(1, s)-inequality such thatX * has the weak* asymptotic-norming property I with respect toB X .  相似文献   

2.
A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, yX. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (xn) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (xn) be aT-martingale taking on values inH. If then x n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (xn) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X * of norm 1 such that If, in addition, the spaceX is strictly convex, x n /n converges weakly; and if the norm ofX * is Fréchet differentiable (away from zero), x n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093  相似文献   

3.
This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.  相似文献   

4.
We characterize locally convex topological algebrasA satisfying: a sequence (x n) inA converges to 0 if, and only if, (x n 2) converges to 0. We also show that a real Banach algebra such thatx n 2+y n 2→0 if, and only if,x n → 0 andy n → 0, for every sequences (x n) and (y n) inA, is isomorphic to, whereX is a compact space.   相似文献   

5.
Summary LetX be a Banach space and for each realt letG(t) andh(t) respectively denote a bounded linear operator onX and a vector inX. Let forx inX. A sufficient condition is given for the existence of a unique vectorx which minimizes μ. An application to control theory is given.
Riassunto SiaX uno spazio di Banach e per ogni realet sianoG(t) eh(t) rispettivamente un operatore lineare limitato suX e un vettore inX. Sia perx inX. Si dà una condizione sufficiente per l'esistenza di un unico vettorex che minimizza μ. Si dà una applicazione alla teoria dei controlli.


Partially supported by N.S.F. grant GP-38596.  相似文献   

6.
Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x * fx*B(X*)}, whereB(X*) is the unit ball inX *.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996.  相似文献   

7.
We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach spaceX is reflexive if and only if it admits and equivalent norm such that there existsx 0≠0 inX and a weak-*-open subsetA of the dual space, satisfying thatx *x 0 attains its numerical radius. for eachx * inA. The first and third author were supported in part by D.G.E.S., project no. BFM 2000-1467. The second author was partially supported by Junta de Andalucía Grant FQM0199.  相似文献   

8.
LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ n * } n=1 ?X* which arew* converging to 0 while inf n x* n ‖>0, are constructed.  相似文献   

9.
Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x1, , xn} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.  相似文献   

10.
We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon-Nikodým property or is Asplund, satisfies n(2)(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon-Nikodým property which satisfy n(2)(X)=1 are the spaces . Also, the only Asplund complex space X with absolute norm which satisfies n(2)(X)=1 is c0(Λ).  相似文献   

11.
In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT onY, and the spectrum ofT is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT onXX, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .  相似文献   

12.
In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l 1(τ) embeds in a Banach spaceX if and only ifL 1([0, 1]τ) embeds inX *. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ1-spaces. It is shown that ifZ * is a dual Banach space, isomorphic to a complemented subspace of anL 1-space, and κ is the density character ofZ *, thenl 1(κ) embeds inZ *. A corollary of this result is that every injective bidual Banach space is isomorphic tol (κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω1, but which is such thatl 11) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω1. The dual space ℰ(S)* is isometric to , and is a member of a new isomorphism class of dualL 1-spaces.  相似文献   

13.
A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (xn) → x in X and (xn*) → 0 in X* with ||xn|| = ||x||= 1 we have (xn*(xn)) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any hH, the evaluation operators from M to H given by SSh, SSth are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.  相似文献   

14.
The multiplicative spectrum of a complex Banach space X is the class (X) of all (automatically compact and Hausdorff) topological spaces appearing as spectra of Banach algebras (X, *) for all possible continuous multiplications on X turning X into a commutative associative complex algebra with unity. Properties of multiplicative spectra are studied. In particular, we show that (X n ) consists of countable compact spaces with at most n nonisolated points for any separable, hereditarily indecomposable Banach space X. We prove that (C[0, 1]) coincides with the class of all metrizable compact spaces. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.  相似文献   

15.
The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x 1,x 2, …,x n inE with ‖x 1‖=‖x 2‖=…=‖x n‖=1, there exists anx inE with ‖x‖=1 such that The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL 1-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ N equipped with the usual 1-norm.  相似文献   

16.
<Emphasis Type="Italic">q</Emphasis>-Besselian Frames in Banach Spaces   总被引:1,自引:0,他引:1  
In this paper, we introduce the concepts of q-Besselian frame and (p, σ)-near Riesz basis in a Banach space, where a is a finite subset of positive integers and 1/p+1/q = 1 with p 〉 1, q 〉 1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and (p, σ)-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X^* that if {xn}n=1^∞ C X is a q-Besselian frame for X, then there exists a p-Besselian frame {y&*}n=1^∞ belong to X^* for X^* such that x = ∑n=1^∞ yn^*(x)xn for all x ∈ X, and x^* =∑n=1^∞ x^*(xn)yn^* for all x^* ∈ X^*. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces.  相似文献   

17.
LetB be a separable Banach space and let {:||1} denote the unit ball ofB *. LetX be a symmetricp-stableB-valued random variable and let {X j } j=1 n be i.i.d. copies ofX. LetB 1 be a finite-dimensional Banach space with a symmetric unconditional basis {y j } j=1 n . An upper bound is obtained for that improves the one given by Giné, Marcus and Zinn [J. Functional Anal. 63, 47–73 (1985)].  相似文献   

18.
LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR n:X→X such thatR nRm=Rmin(n,m) ifn≠m and lim n→∞ R n x=x for allx∈X. We prove that, ifR n−Rn −1 factors uniformly through somel p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .  相似文献   

19.
Let (t n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t n+1?t n α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors xX and x ?X ?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.  相似文献   

20.
We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ1, λ2, . . . , λ n , and the operators T and r j (T − λ j I)−1 are polynomially bounded, then there exists a nontrivial common invariant subspace for T * and (T − λ j I)*-1.  相似文献   

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