Banach Sequence Spaces Research Articles

Almost overcomplete and almost overtotal sequences in Banach spaces II

A sequence in a separable Banach space X 〈resp. in the dual space X⁎〉 is said to be overcomplete (OC in short) 〈resp. overtotal (OT in short) on X〉 whenever the linear span of each subsequence is dense in X 〈resp. each subsequence is total on X〉. A sequence in a separable Banach space X 〈resp. in the dual space X⁎〉 is said to be almost overcomplete (AOC in short) 〈resp. almost overtotal (AOT in short) on X〉 whenever the closed linear span of each subsequence has finite codimension in X 〈resp. the annihilator (in X) of each subsequence has finite dimension〉. We provide information about the structure of such sequences. In particular it can happen that, an AOC 〈resp. AOT〉 given sequence admits countably many not nested subsequences such that the only subspace contained in the closed linear span of every of such subsequences is the trivial one 〈resp. the closure of the linear span of the union of the annihilators in X of such subsequences is the whole X〉. Moreover, any AOC sequence {xn}n∈N contains some subsequence {xnj}j∈N that is OC in [{xnj}j∈N]; any AOT sequence {fn}n∈N contains some subsequence {fnj}j∈N that is OT on any subspace of X complemented to {fnj}j∈N⊤.

On the defect of compactness in Banach spaces

Concentration compactness methods improve convergence for bounded sequences in Banach spaces beyond the weak-star convergence provided by the Banach–Alaoglu theorem. A further improvement of convergence, known as profile decomposition, is possible up to defect of compactness, a series of “elementary concentrations” defined relative to the action of some group of linear isometric operators. This note presents a general profile decomposition for uniformly convex and uniformly smooth Banach spaces, generalizing the result of one of the authors (S.S.) for Sobolev spaces and of the other (C.T. jointly with I. Schindler) for general Hilbert spaces. Unlike in the Hilbert space case, profile decomposition is based not on weak convergence, but on a different mode of convergence, called polar convergence, which coincides with weak convergence if and only if the norm satisfies the known Opial condition, used in the context of fixed point theory for nonexpansive maps.

SPECTRUM AND COMPACTNESS OF THE CESÀRO OPERATOR ON WEIGHTED SPACES

An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1<p<\infty$, for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the weighted setting (for example, existence of eigenvalues, compactness) which are not present in $\ell _{p}$.

The Cesàro Operator and Unconditional Taylor Series in Hardy Spaces

We introduce the spaces \({H^{p}_{uc}}\) consisting of all functions in the Hardy space \({H^p, 1 < p < \infty}\), whose Taylor series are unconditionally convergent and analyze the action of the Cesaro operator in these spaces. There is a related class of Banach sequence spaces \({N^p, 1 < p < \infty}\), arising from harmonic analysis, in which the discrete Cesaro operator acts. The classical majorant property (due to Hardy and Littlewood) provides a means to transfer various results about the Cesaro operator in Np (e.g. continuity, spectrum, etc.) to those for the corresponding Cesaro operator acting in \({H^{p}_{uc}. \,\,{\rm For}\,\, p \neq 2}\), the space \({H^{p}_{uc}}\) is rather different to the classical space \({H^{p}}\). The spaces \({N^{p}}\) also exhibit a remarkable stability property under averaging, akin to that established by Bennett for \({\ell^{p}}\).

On Some Banach Sequence Spaces Derived by a New Band Matrix

In this paper, we define the band matrix T = (tnk) by tnk =  tn , k = n − 1 tn , k = n− 1 0 , k > n or 0 ≤ k 0 for all n ∈ N and (tn) ∈ c\c0. By using the matrix T , we introduce the sequence space `p(T ) for 1 ≤ p ≤ ∞. Also, we give some inclusion theorems related to this space and find the α-, β-, γduals of the space `p(T ). Lastly, we investigate the necessary and sufficient conditions for an infinite matrix to be in the classes (`p(T ), λ) or (λ, `p(T )) and give the norm of the operators in B(`p(T ), μ(S)), where λ ∈ {`1, c0, c, `∞} and μ ∈ {`1, `∞}.

Function Spaces and Applications

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

Almost overcomplete and almost overtotal sequences in Banach spaces

We introduce the new concepts of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we present several applications of this fact.

FRÉCHET FRAMES, GENERAL DEFINITION AND EXPANSIONS

We define an (X1, Θ, X2)-frame with Banach spaces X2⊆ X1, ‖ ⋅ ‖1≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XFand its dual [Formula: see text], as well of some of the generating spaces of XFwith convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF→ ΘF. Several examples illustrate our investigations.

Extendibility of bilinear forms on banach sequence spaces

We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize c 0 in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.

Some combinatorial principles for trees and applications to tree families in Banach spaces

Suppose that is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences .

Compact groups of positive operators on Banach lattices

In this paper, we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.

Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.

Application of Measure of Noncompactness to Infinite Systems of Differential Equations

AbstractIn this paper we determine theHausdorff measure of noncompactness on the sequence space n(ϕ) ofW. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces n(ϕ) and m(ϕ). Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

Mean separations in Banach spaces under abstract interpolation and extrapolation

We introduce the mean separation for bounded sequences in Banach spaces and the related seminorm for bounded linear operators. The introduced quantities are closely related to the geometric characterizations of the Banach–Saks property and the alternate signs Banach–Saks property. We investigate the behavior of the mean separations for a class of operators between vector-valued Banach sequence spaces E(Xν), providing that a Banach sequence lattice E has the Banach–Saks property. We estimate the mean separations for operators under abstract interpolation and extrapolation methods. In particular, we obtain quantitative and qualitative results on the heredity of the Banach–Saks properties under these methods.

Some topological and geometrical properties of new Banach sequence spaces

In the present paper, we introduce a new band matrix and define the sequence space where is the k th Fibonacci number for every . We also establish some inclusion relations concerning this space and determine its α-, β-, γ-duals. Further, we characterize some matrix classes on the space and examine some geometric properties of this space. MSC:11B39, 46A45, 46B45, 46B20.

Perturbations of Generalized Dual Banach Frames

The classical duals and generalized duals of frames play a fundamental role in the abstract frame theory. In this paper we first define the concepts of dual, canonical dual, pseudo-dual and approximate dual for Bessel sequences with respect to a BK-space of scalar-valued sequences in Banach spaces, and illuminate their relationship with Banach frames. Then we prove that classical dual and approximate dual of Banach frames are stable under small perturbations so that the results obtained1 is a special case of it. We also apply a new characterization of classical dual Banach frames to discuss a stability problem for them. For approximate dual Banach frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction.

Perturbations of Generalized Dual Banach Frames

The classical duals and generalized duals of frames play a fundamental role in the abstract frame theory. In this paper we first define the concepts of dual, canonical dual, pseudo-dual and approximate dual for Bessel sequences with respect to a BK-space of scalar-valued sequences in Banach spaces, and illuminate their relationship with Banach frames. Then we prove that classical dual and approximate dual of Banach frames are stable under small perturbations so that the results obtained1 is a special case of it. We also apply a new characterization of classical dual Banach frames to discuss a stability problem for them. For approximate dual Banach frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction.

A Note on Generalized Approximation Property

We introduce a notion of generalized approximation property, which we refer to as --AP possessed by a Banach space , corresponding to an arbitrary Banach sequence space and a convex subset of , the class of bounded linear operators on . This property includes approximation property studied by Grothendieck, -approximation property considered by Sinha and Karn and Delgado et al., and also approximation property studied by Lissitsin et al. We characterize a Banach space having --AP with the help of -compact operators, -nuclear operators, and quasi--nuclear operators. A particular case for () has also been characterized.

Matrix Mappings on the Domains of Invertible Matrices

We focus on sequence spaces which are matrix domains of Banach sequence spaces. We show that the characterization of a random matrix operator , where and are matrix domains with invertible matrices and , can be reduced to the characterization of the operator . As an application, the necessary and sufficient conditions for the matrix operators between invertible matrix domains of the classical sequence spaces and norms of these operators are given.

Frames and Riesz bases for Banach spaces, and Banach spaces of vector-valued sequences

This paper is devoted to an investigation of frames and Riesz bases for general Banach sequence spaces. We establish various relationships between Bessel (respectively, frames) and Riesz sequences (respectively, Riesz bases), and then some of their applications are presented. Some recent results for Banach frames and atomic decompositions are sharpened with simple proofs. Banach spaces consisting of Bessel or Riesz sequences are introduced and it is shown that they are isometrically isomorphic to some Banach spaces of bounded linear operators, and that some subspaces of those Banach spaces are isometrically isomorphic to some Banach spaces of compact operators.