Symmetric Sequence Spaces Research Articles

The Schur and (weak) Dunford-Pettis properties in Banach lattices

AbstractWe study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for 1 ≤ u ≤ 2 1 \le u \le 2 the identity map id: ℓ u ↪ ℓ 2 \ell _u \hookrightarrow \ell _2 is absolutely ( u , 1 ) (u,1) -summing, i. e., for every unconditionally summable sequence ( x n ) (x_n) in ℓ u \ell _u the scalar sequence ( ‖ x n ‖ ℓ 2 ) (\|x_n \|_{\ell _2}) is contained in ℓ u \ell _u , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2 2 -concave symmetric Banach sequence space E E the identity map id : E ↪ ℓ 2 \text {id}: E \hookrightarrow \ell _2 is absolutely ( E , 1 ) (E,1) -summing, i. e., for every unconditionally summable sequence ( x n ) (x_n) in E E the scalar sequence ( ‖ x n ‖ ℓ 2 ) (\|x_n \|_{\ell _2}) is contained in E E . Various applications are given, e. g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator T T on ℓ 2 \ell _2 with values in a 2 2 -concave symmetric Banach sequence space E E is a multiplier from ℓ 2 \ell _2 into E E . Furthermore, we prove an asymptotic formula for the k k -th approximation number of the identity map id : ℓ 2 n ↪ E n \text {id}: \ell _2^n \hookrightarrow E_n , where E n E_n denotes the linear span of the first n n standard unit vectors in E E , and apply it to Lorentz and Orlicz sequence spaces.

Q-concavity and related properties on symmetric sequence spaces

We introduce a new property between the q-concavity and the lower q-estimate of a Banach lattice and we get a general method to construct maximal symmetric sequence spaces that satisfies this new property but fails to be q-concave. In particular this gives examples of spaces with the Orlicz property but without cotype 2.

Q-CONCAVITY AND q-ORLICZ PROPERTY ON SYMMETRIC SEQUENCE SPACES

We give a general method for constructing symmetric sequence spaces that for $1 \lt q \lt 2$ satisfy a lower q-estimate but fail to be q-concave and, for $2 · q \lt 1$, have the q-Orlicz property but fail to be q-concave. In particular, this gives examples of spaces with the 2-Orlicz property but without cotype 2.

The Dunford-Pettis Property for Symmetric Spaces

AbstractA complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that ℓ1, c0 and ℓ∞ are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on (0, α), 0 < α ≤ ∞, the only spaces with the Dunford-Pettis property are L1, L∞, L1 ∩ L∞, L1 + L∞, (L∞)◦ and (L1 + L∞)◦, where X◦ denotes the norm closure of L1 ∩ L∞ in X. It is also proved that all Banach dual spaces of L1 ∩ L∞ and L1 + L∞ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces (L1 + L∞)◦ and (L∞)◦ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Köthe-Bochner spaces.

Symmetric Sequence Subspaces of C (α)

Ifis an ordinal, then the space of all ordinals less than or equal tois a compact Hausdorff space when endowed with the order topology. Let C(�) be the space of all continuous real-valued functions defined on the ordinal interval (0,�). We characterize the symmetric sequence spaces which embed into C(�) for some countable ordinal �. A hierarchy (E�) of symmetric sequence spaces is constructed so that, for each countable ordinal �, Eembeds into C(! ! � ), but does not embed into C(! ! � ) for any � < �.

Symmetric sequence subspaces of C(α), II

AbstractIf α is an ordinal, then the space of all ordinals less than or equal to α is a compact Hausdorff space when endowed with the order topology. Let C(α) be the space of all continuous real-valued functions defined on the ordinal interval [0; α]. We characterize the symmetric sequence spaces which embed into C(α) for some countable ordinal α. A hierarchy (Eα) of symmetric sequence spaces is constructed so that, for each countable ordinal α, Eα embeds into C(ωωα), but does not embed into C(ωωβ) for any β &lt; α.

Orlicz property and cotype in symmetric sequence spaces

We construct a symmetric sequence space that satisfies the Orlicz property but that fails to be of cotype 2.

Local uniform and uniform convexity of non-commutative symmetric spaces of measurable operators

Let E be a separable symmetric sequence space, and let CE be the unitary matrix space associated with E, i.e. the Banach space of all compact operators x on l2 so that s(x) E, with the norm , where are the s-numbers of x. One of the interesting subjects in the theory of the unitary matrix spaces is the clarification of correlation between the geometric properties of the spaces E and CE. A series of results in this direction related with the notions of type, cotype and uniform convexity of the spaces CE has been already obtained (see 13).

Symmetric sequence spaces, bases, and applications

Symmetric sequence spaces, bases, and applications

Isometries of complex symmetric sequence spaces

Isometries of complex symmetric sequence spaces

Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces

For every symmetric sequence space E we denote by C E the associated unitary matrix space. We show that every basic sequence in C E has a subsequence which embeds in l 2 ⊕ E. This reduces many problems about C E to the analogous problems on E. We also study the asymptotic behavior of the embeddings of one unitary sequence space into another, and the problem of the uniqueness of the symmetric structure of unitary matrix spaces.

More on convergence in unitary matrix spaces

Let E E be a symmetric sequence space satisfying the Radon-Riesz Property \[ { | | x n | | → | | x | | and x n → x weakly } ⇒ | | x n − x | | → 0 , \{ ||{x_n}|| \to ||x||{\text {and }}{x_n} \to x{\text { weakly}}\} \Rightarrow ||{x_n} - x|| \to 0, \] then the same is true for the associated unitary matrix space C E C_{E} .

Schauder Bases and Norm Ideals of Compact Operators

This paper studies the relationship between unitarily invariant crossnorms on the tensor product of Hilbert spaces and the corresponding symmetric sequence spaces. It is shown that problems concerning the behavior of norm ideals of compact operators on Hilbert space can often be treated successfully by translating the problem to one concerned with basis theory (and hence, perhaps, to one more transparent). For example, Schatten has asked for necessary and sufficient conditions that the conjugate space of a minimal norm ideal again be a minimal norm ideal. Using the ideas developed here we give such a characterization.

Schauder bases and norm ideals of compact operators

This paper studies the relationship between unitarily invariant crossnorms on the tensor product of Hilbert spaces and the corresponding symmetric sequence spaces. It is shown that problems concerning the behavior of norm ideals of compact operators on Hilbert space can often be treated successfully by translating the problem to one concerned with basis theory (and hence, perhaps, to one more transparent). For example, Schatten has asked for necessary and sufficient conditions that the conjugate space of a minimal norm ideal again be a minimal norm ideal. Using the ideas developed here we give such a characterization.

Absolutely ?-summing operators, ? a symmetric sequence space

Pietsch [5] introduced the concept of absolutely summing operators in Banach spaces and later in [6] extended this concept to absolutely p-summing operators. At the background of these concepts are the sequence spaces I p and their duality theory. The object of the present paper is to extend the above concept to abstract sequence spaces 2. The sequence spaces 2 involved are described in Section 2; the absolutely 2-summing operators are studied in Section3 while Section4 discusses the interesting special case 2=n(~b), a sequence space which includes for special q5 the P and l ~ spaces and was introduced in the literature by Sargent [10].

A Class of Reflexive Symmetric Bk-Spaces

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A sequence space E is a BK-space (9) if it is equipped with a norm under which: first, E is a Banach space and second, each of the coordinate maps x → xi is continuous. Let ∑ be the group of all permutations of Z+ = {1, 2, 3, …}. If x ∈ ω and σ ∈ ∑, the sequence xσ is defined by (xσ)i = xσ(i)). A sequence space E is symmetric if xσ ∈ E whenever x ∈ E and σ ∈ ∑. Accounts of symmetric sequence spaces occur in (3; 7; 8).

On Symmetric Sequence Spaces

Proceedings of the London Mathematical SocietyVolume s3-16, Issue 1 p. 85-106 Articles On Symmetric Sequence Spaces D. J. H. Garling, D. J. H. Garling St John's College, CambridgeSearch for more papers by this author D. J. H. Garling, D. J. H. Garling St John's College, CambridgeSearch for more papers by this author First published: 1966 https://doi.org/10.1112/plms/s3-16.1.85Citations: 34AboutPDF 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 onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-16, Issue11966Pages 85-106 RelatedInformation