Banach Sequence Spaces Research Articles

Asymptotic Behaviour of Unbounded Nonexpansive Sequences in Banach Spaces

Asymptotic Behaviour of Unbounded Nonexpansive Sequences in Banach Spaces

Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces

Let X be a real Banach space, (x0 )n>O a nonexpansive sequence in X (i.e., llxil xj+1 11 0), and C the closed convex hull of the sequence (x,+i x0),>o We prove that lim0+OO llxn/nll = infn>l II(xn xo)/nll = infZEC lizil and deduce a simple short proof for the following result. (i) If X is reflexive and strictly convex, then x,,/n converges weakly in X to the element of minimum norm PCO in C with IIPCOII = inf Xn -XO = lim Xn n>l fl n-'+0O fl (ii) If X* has Frechet differentiable norm, then X/nl converges strongly to PCO. This result contains previous results by Pazy, Kohlberg and Neyman, Plant and Reich, and Reich and is also optimal since the assumptions made on X in (i) or (ii) are also necessary for the respective conclusion to hold.

Upper lp-estimates in vector sequence spaces, with some applications

In [11], Partington proved that if λ is a Banach sequence space with a monotone basis having the Banach-Saks property, and (Xn) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space ΣλXn has this same property. In addition, Partington gave an example showing that if λ and each Xn, have the weak Banach-Saks property, then ΣλXn need not have the weak Banach-Saks property.

Guerre‐Delabrière, Syl Vie: Classical sequences in Banach spaces. Marcel Dekker, Inc., New York‐Basel‐Hong Kong 1992, XIV, 207 pp., $ 99.75

Biometrical JournalVolume 35, Issue 7 p. 800-800 Book Review Guerre-Delabrière, Syl Vie: Classical sequences in Banach spaces. Marcel Dekker, Inc., New York-Basel-Hong Kong 1992, XIV, 207 pp., $ 99.75 Werner Linde, Werner LindeSearch for more papers by this author Werner Linde, Werner LindeSearch for more papers by this author First published: 1993 https://doi.org/10.1002/bimj.4710350706AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume35, Issue71993Pages 800-800 RelatedInformation

Quelques espaces fonctionnels associés à des processus gaussiens

The first part of the paper presents results on Gaussian measures supported by general Banach sequence spaces and by particular spaces of Besov-Orlicz type. In the second part, a new constructive isomorphism between the just mentioned sequence spaces and

UPPER BOUNDS AND APPROXIMATION FOR THE SOLUTION OF A NONLINEAR PARTICLE TRANSPORT PROBLEM WITH MAXWELL TYPE BOUNDARY CONDITIONS

We consider a nonlinear particle transport problem in a slab of thickness 2a with Maxwell-type boundary conditions. The nonlinearity is due to the presence of a scattering term, which is a quadratic function of the (unknown) particle density u(t)=u(x, y; t). We first prove that, under suitable assumptions, the integral version of the transport equation has a unique strongly continuous solution u=u(t)∈L1∩L∞, defined over an arbitrary fixed time interval [0, t0]. We also evaluate upper bounds for the norms ||u(t)|| and ||u(t)||∞, which are useful to understand the mathematical and the physical properties of the particle transport problem under consideration. Finally, we discretize the angle variable y in the unknown function u(t)=u(x, y; t), (i.e., we use the “discrete ordinate method”) and we prove the convergence of the discretized solution to the exact solution by using a generalization of Trotter’s theory on sequences of Banach spaces approximating a given Banach space.

A non-linear method for constructing certain basic sequences in Banach spaces

isomorphic to an infinite dimensional conjugate space. This is, of course, a first step toward the better known conjecture that every Banach space contains an isomorphic copy of c o or or an infinite dimensional reflexive subspace. In this paper, we exhibit a new technique for constructing infinite boundedly complete basic sequences and consequently separable conjugate subspaces. Aninteresting feature of this method is that it involves a non-linear approach, which is in sharp contrast with the linear nature of the problem we address. Our approach also combines techniques originating in the local theory of Banach spaces (Dvoretzky’s theorem and concentration phenomenon) with infinite dimensional concepts like dentability and transfinite slicing of sets. One aPllication of this method is that Banach spaces with the Analytic Radon-Nikbdym Property (ARNP) contain copies of infinite dimensional conjugate spaces. (Recall that a complex Banach space X is said to have the ARNP if every X-valued bounded analytic map on the open unit disc of the complex plane has radial limits almost surely). This class of spaces containsbesides those possessing the Radon-Nikodym property (RNP) (see [D-U])--ali Banach lattices not containing c o [B-D] as well as all preduals of Von Neuman algebras [H-P]. What is needed for the proof is the following geometric characterization of such spaces established in [G-L-M]: Every bounded subset of a Banach space with the ARNP has arbitrarily norm-small determined by Lipschitz and plurisubharmonic functions. In the classical RNP setting (where the are determined by continuous linear functionals) the analogous statement (i.e. The existence of infinite dimensional conjugate spaces) was established in [G-M1]. Also shown there is the case where the slices are determined by a finite number of linear functionals i.e.

Representing types in Orlicz and Lorentz sequence spaces

AbstractThe aim of this paper is to give a representation for the types in some stable Banach sequence spaces, namely in the Orlicz, Lorentz and dual of Lorentz sequence spaces. We also find a characterization for the Lorentz sequence spaces whose class of weakly-null types is locally compact for the topology of uniform convergence on bounded subsets.

On seeking buckling modes of a circular plate

A construction of the buckling modes of a circular plate is examined by using the solutions of an infinite system of non-linear algebraic equations that appears on substituting their non-trivial solution representable, by assumption, in the form of series, into the non-linear Karman equations. It is shown that an approximation can be found to the solution of the system by using a projection method and a projection-iteration process in the Banach space of sequences whose series from the elements converge absolutely. Results of computations are presented.

An infinite-dimensional extension of theorems of Hartman and Wintner on monotone positive solutions of ordinary differential equations

Consider the equation (1) x + A(t)x = -,f(t, x) x(O) = x?, x? C X, a Banach sequence space with a Schauder Basis. It is proved that if f(t, 0) = 0, 4(t)(') + f(t, ) is a positive operator and the solution operator K(t, O)x = x fJ A(s) ds fS f(s, x(s)) ds is compact for t > 0, then system (1) has at least one solution x(t) , x(t) O such that x(t) > 0, *(t) 0 .

Example of a sequentially incomplete regular inductive limit of Banach spaces

A sequentially incomplete regular inductive limit of a sequence of Banach spaces is constructed.

Inductive limits of vector-valued sequence spaces

Let L be a normal Banach sequence space such that every element in L is the limit of its sections and let E = ind En be a separated inductive limit of the locally convex spaces. Then ind L(En) is a topological subspace of L(E).

Onc 0 sequences in Banach spaces

A Banach space has property (S) if every normalized weakly null sequence contains a subsequences equivalent to the unit vector basis ofc0. We show that the equivalence constant can be chosen “uniformly”, i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture that a separable Banach space with property (u) not containingl1 has a separable dual.

Hermitian operators and isometries on sums of Banach spaces

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

Fréchet spaces of Moscatelly type

A certain class of Frechet tpaces, called of Moscatelli type, it introduced and studied. tising sorne shifiing device these Fr?chet spaces are defined as projective limits of Hanach spaces L«XDSE N), where L isa normal Banach tequence space and the X~’s are Banach tpaces. The duality between Frechet and (LB)-spacet of Moscatelli type it eslablished and the following properties of Frechet tpacet are characterized in the present context: distinguishedness, quasinonnability, Heinrich’s density condition, existence of a continuous noran in the space Gr the bidual, and the properties (DN) and (O) of Vogt. Tite aim of titis anide is to study a class of Frecitet spaces witicit itas been used recently quite obten to find counterexamples titat solved several open questions in tite titeory ob Frecitet and (DF)-spaces. In 1980, Moscatelli [13] introduced a certain type of Frecitet and (LB)spaces to find a twisted quojection, i.e., a Frecitet space witicit is a surjective limit of Banacit spaces, wititout a continuous norm but not isomorpitic to a product of Frecitet spaces eacit itaving a continuous norm. Sucit a space cannot itave an unconditional basis, according to [7]. Tite natural extension of tite classieal idea of Moscatelli yields tite following construction. We start with a normal Banach sequence space (L, II II), two sequences of Banach spaces (XJ5,,. (flPc.N and linear continuous maps f, Y5 —*2 ’, and 80 1 Bonet-S. Dierolf foreveryice Rl lei f& Yk—*Xk be a /inearmapsucir thatf,(Bk)cAk. Foreveryn e Rl, tire space F.,:=L((Yk,sk), .,) is a Banacir space according lo 1.2, and tire linear inap g,:F.,~1—*F.,, (zjk,N—*((zD,.C,,, f.,(z,,), (zk)k>,) is non decreasing. Tire Frechel space ofMoscaie//i iype associated lo (or wfth respect lo-w. rl.) (L,ii II), (Xkrjk,N, ~ andf,Y,.,—>X?icE Rl) is tire Freciret space dejinedby F= proj ((F9,,,~,(g,,).,~~) i.e., tire projective /imit of tire projective sequence of fianacir spaces (F.,).,,~ with /inking niaps (g,,),,~. 14. Proposition. Tire Frecirel space ofMoscatelil iype defined aboye coincides a/gebraically with {y=(yD,~N eflY, ’ w.ni. tire inc/usion j:F.—*f1(Y~,s~) and tire /inear niap kc N J?F—*L((Xk,rk)kEN). fty):=(f(yj)5,~. Proof. F coincides witit tite space ((zOk.Ne 1117 : gjzJ~’)=z’for al/le Rl>, enjcN dowed witit tite topology induced by [117. We denote by H tite space JEN (ye ~ : ~&k))k.~ E L((XS,rD,,N)> endowed witit tite initial topology w.r.t. tite ker’J linear mapsj and ji Define ~rF—4’Hby w((zJ)~CN).~(zk+L) . It isa direct matter to citeck titat i}i is linear, bijective and continuous (see [4,(l.3)(3)]). Since F and H are botit Frecitet spaces, w is also open tite proof is complete. a From now on we sitali make tite identificationindicated in Proposition 1.4. We close titis section recalling tite definition ob (LB)-spaccs ob Moscatelli type and some of titeir properties from [4]. Let (L,II II) be a normal Banacit sequence space. Let ,,) is a Banacit space, nc Rl. Tite corresponding(LB)-space of Moscatelil type is defined by E: = md E.,. Tite closed unit balI of E,, wifl be denoted by JS,. A basis ob 0-neigbbouritoods (0-ngitbs) in E is given by tite sets of tite bomi @s,,A,-~-8~%, 8>0, 45’ 0(ke Rl). N If(L,li II) satisfies property (‘y), titen E is regular it’ and only ib E Rl ~ 1 Vic~n B,,cr pR,, witere B,, denotes tite closure of fi,, in (X,,,r,,). In [4,Section 3] we associated to E a projective limit in tite following way. Qiven 8>0 and EK>O(ice Rl), tite Minkowski bunctional of s,,A,+BB, is denotad by p~8, and it is a norm on X,< equivalent to r,,. Titen E is tite projective limit Frecher spaces ofMoscalelil type 81 Ch L(<XbPEDSFN). 5(8~) Tite (LB)-space E is continuously injected in E and E is a complete (DF)space. A basis of 0-nghbs in E is given by tite sets

Lifting results for sequences in Banach spaces

Several important classes of Banach spaces are characterized by means of convergence properties of sequences. For example, if X is a Banach space, then X belongs to the class Nl1 of spaces without copies of l1, the class R of reflexive spaces or the class F of finite-dimensional spaces if and only if each bounded sequence has respectively a weakly Cauchy (w-Cauchy), weakly convergent (w-convergent) or convergent subsequence. Similarly X is in the class WSC of weakly sequentially complete spaces, or the class SCH of spaces with the Schur property if and only if each w-Cauchy sequence is w-convergent, or convergent, respectively; note that X ∈ SCH if and only if each w-convergent sequence of X is convergent (see [12], p. 47).

Point spectra of Cesàro matrices

For all positive a the point spectrum of the ( C, α) matrix is determined, where the matrix is regarded as an operator on certain Banach sequence spaces. In particular the point spectrum is obtained in the spaces I p ( X), with 1< p≤∞, where X is a Banach space.

The regularity of Dunford-Pettis operators

Let λ \lambda denote a symmetric, solid Banach sequence space having { e i } i = 1 ∞ \left \{ {{e_i}} \right \}_{i = 1}^\infty as a symmetric basis and considered as a Banach lattice with order defined coordinatewise. A complete description of the relationship between regular and Dunford-Pettis operators T : L 1 [ 0 , 1 ] → λ T:{L^1}[0,1] \to \lambda is given. The results obtained complete earlier work of Gretsky and Ostroy and of the author in this area.

LU-factorization of order bounded operators on Banach sequence spaces

LU-factorization of order bounded operators on Banach sequence spaces

On a classification of sequences in Banach spaces

On a classification of sequences in Banach spaces