On a class of Köthe sequence spaces with normal structure

In this note, we investigate the generalized modulus of convexity δ ( λ ) and the generalized modulus smoothness ρ ( λ ) . We obtain some estimates of these moduli for X = lp . We obtain inequalities between WCS coefficient of a K¨othe sequence space X and δ ( λ ) X . We show that, for a wide class of K¨othe sequence spaces X, if for some ε∈ (0, 9 10 ] holds δ X (ε) > 1 3 1 √ 3 2 ε, then X has normal structure.     

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article,we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology,such as small covers,quasi-toric manifolds and(real)moment-angle manifolds;especially for the cases of small covers and quasi-toric manifolds.These kinds of orbit configuration spaces have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope.As a by-product of our method,we also obtain a formula of the Euler characteristic for the classical configuration space,which generalizes the Félix-Thomas formula.In addition,we also study the homotopy type of such orbit configuration spaces.In particular,we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.     

Some <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces defined by Orlicz functions

In this article we introduce the sequence spaces cI(M),c0I(M),mI(M) and m0I(M) using the Orlicz function M.We study some of the properties like solid,symmetric,sequence algebra,etc and prove some inclusion relations.     

THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES

The functional dimension of countable Hilbert spaces has been discussed by some authors. They showed that every countable Hilbert space with finite functional dimension is nuclear. In this paper the authors do further research on the functional dimension, and obtain the following results: (1) They construct a countable Hilbert space, which is nuclear, but its functional dimension is infinite. (2) The functional dimension of a Banach space is finite if and only if this space is finite dimensional. (3) Let B be a Banach space, B* be its dual, and denote the weak * topology of B* by σ(B*,B). Then the functional dimension of (B*,σ(B*,B)) is 1. By the third result, a class of topological linear spaces with finite functional dimension is presented.     

Distributional Dimension of Fraetal Sets in Local Fields

The distributional dimension of fractal sets in R^n has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions. Keywords local field, B-type space, F-type space, distributional dimension, Hausdorff dimension Fourier dimension     

ALMOST CONVERGENCE AND DOUBLE SEQUENTIAL BAND MATRIX

The class f of almost convergent sequences was introduced by G.G. Lorentz,using the idea of the anach limits [A contribution to the theory of divergent sequences,Acta Math. 80(1948), 167–190]. Let f0( ) and f( ) be the domain of the double sequential band matrix ( r, s) in the sequence spaces f0 and f. In this article, the β-and γ-duals of the space f( ) are determined. Additionally, we give some inclusion theorems concerning with the spaces f0( ) and f( ). Moreover, the classes(f( ) : μ) and(μ : f( )) of infinite matrices are characterized, and the characterizations of some other classes are also given as an application of those main results, where μ is an arbitrary sequence space.     

DOMAIN OF THE DOUBLE SEQUENTIAL BAND MATRIX B（r,s） ON SOME MADDOX＇S SPACES

The domain of generalized difference matrix B(r, s) in the classical spaces l∞,c, and c0 was recently studied by Kirisci and Bassar in [16]. The main goal of this article is to introduce the paranormed sequence spaces l∞( B, p), c( B, p), and c0( B, p), which are more general and comprehensive than the corresponding consequences of the matrix domain of B(r, s), as well as other studies in literature. Besides this, the alpha-, beta-, and gamma-duals of the spaces l∞( B, p), c( B, p), and c0( B, p) are computed and the bases of the spaces c( B, p)and c0( B, p) are constructed. The final section of this article is devoted to the characterization of the classes(λ( B, p) :) and( : λ( B, p)), where λ∈ {c, c0, l∞}and is any given sequence space. Additionally, the characterization of some other classes which are related to the space of almost convergent sequences is obtained by means of a given lemma.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

THE SPACES OF CESARO ALMOST CONVERGENT SEQUENCES AND CORE THEOREMB

As known,the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces.However,the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new(see [15]).The purpose of this paper is to introduce the new spaces f and f0 consisting of all sequences whose Cesa`ro transforms of order one are in the spaces f and f0,respectively.Also,in this paper,we show that f and f0 are linearly isomorphic to the spaces f and f0,respectively.The β-and γ-duals of the spaces f and f0 are computed.Furthermore,the classes(f:) and(:f) of infinite matrices are characterized for any given sequence space,and determined the necessary and sufficient conditions on a matrix A to satisfy BC-core(Ax) K-core(x),K-core(Ax) BC-core(x),BC-core(Ax) BC-core(x),BC-core(Ax) st-core(x) for all x ∈∞.     

The role of BMOA in the boundedness of weighted composition operators on the unit ball

We study the weighted composition operators Wh,on Hardy space H2(B) whenever h ∈ BMOA(resp.h ∈ VMOA).Analogous results are given for Hp(B) spaces and the scale of weighted Bergman spaces.In the latter case,BMOA is replaced by the Bloch space(resp.VMOA by the little Bloch space).     

On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function

In this article we introduce the paranormed sequence spaces(f,Λ,△m,p),c0(f,Λ,△m,p) and ■∞(f,Λ,△m,p),associated with the multiplier sequence Λ =(λk),defined by a modulus function f.We study their different properties like solidness,symmetricity,completeness etc.and prove some inclusion results.     

On some new generalized difference sequence spaces on seminormed spaces defined by a sequence of Orlicz functions

In this paper we define the sequence space ℓ _M (Δ _υ ^m , p, q, s) on a seminormed complex linear space, by using a sequence of Orlicz functions. We study some algebraic and topological properties. We prove some inclusion relations involving ℓ _M (Δ _υ ^m , p, q, s). spaces     

On a class of difference sequences related to the ℓ<Superscript>p</Superscript> space defined by Orlicz functions

In this article we introduce the difference sequence space m(M, Δ, φ) using the Orlicz function. We study its different properties like solidity, completeness etc. Also we obtain some inclusion relations involving the space m(M, Δ, φ).      

Generalized vector-valued paranormed sequence space using modulus function

In this paper we introduce a generalized vector-valued paranormed sequence space N_p(E_k,Δ^m,f,s) using modulus function f, where p=(p_k) is a bounded sequence of positive real numbers such that inf_kp_k>0,(E_k,q_k) is a sequence of seminormed spaces with E_k+1⊆E_k for each k ∈ N and s?0. We have also studied sequence space N_p(E_k,Δ^m,f^r,s), where f^r=f°f°f°,…,f (r-times composition of f with itself) and r∈N={1,2,3,…}. Results regarding completeness, K-space, normality, inclusion relations etc. are derived. Further, a study of multiplier of the set N_p(E_k,f,s) is also made by choosing (E_k,‖·‖_k) as sequence of normed algebras.     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

Imbedding of power series spaces and spaces of analytic functions

The diametral dimension of a nuclear Fréchet spaceE, which satisfies (DN) and (Ω), is related to power series spaces Λ₁(ε) and Λ_∞(ε) for some exponent sequence ε. It is proved thatE contains a complemented copy of Λ_∞(ε) provided the diametral dimensions ofE and Λ_∞(ε) are equal and ε is stable. Assuming Λ₁(ε) is nuclear, any subspace of Λ₁(ε) which satisfies (DN), can be imbedded intoE. Applications of these results to spaces of analytic functions are given. Support of Turkish Scientific and Technical Research Council is gratefully acknowledged.     

Strongly almost convergent generalized difference sequences associated with multiplier sequences

Let Λ = (λ _k ) be a sequence of non-zero complex numbers. In this paper we introduce the strongly almost convergent generalized difference sequence spaces associated with multiplier sequences i.e. w ₀[A,Δ ^m ,Λ,p], w ₁[A,Λ ^m ,Λ,p], w _∞[A,Δ ^m ,Λ,p] and study their different properties. We also introduce Δ _Λ ^m -statistically convergent sequences and give some inclusion relations between w ₁[Δ ^m ,λ,p] convergence and Δ _Λ ^m -statistical convergence. Communicated by Pavel Kostyrko     

Calderón--Lozanovskii; construction for mixed norm spaces

We show that the Calderón--Lozanovskii; construction φ(.) commutes with arbitrary mixed norm spaces, that is, φ(E₀[F₀], E₁[F₁]) = φ(E₀, E₁) [φ(F₀, F₁)] if and only if φ is equivalent to a power function. This result we obtain by giving characterizations of the corresponding embeddings of φ(E₀[F₀], E₁[F₁]) into φ₀ (E₀, E₁)[φ₁ (F₀, F₁)] and vice versa in terms of the functions φ, φ₀, φ₁. As a particular case, we get embeddings of an Orlicz space with mixed norms into an Orlicz space on a product of measure spaces. Applications to classical operators between mixed norm Orlicz spaces are also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized difference sequence spaces on seminormed space defined by Orlicz functions

In this paper we define the sequence space ℓ _M (Δ ^m , p, q, s) on a seminormed complex linear space by using an Orlicz function. We study its different algebraic and topological properties like solidness, symmetricity, monotonicity, convergence free etc. We prove some inclusion relations involving ℓ _M (Δ ^m , p, q, s).      

On the Extension of Isometries between Unit Spheres of E and C（Ω）

In this paper,we study the extension of isometries between the unit spheres of some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S1(E) of all smooth points of the unit sphere S1(E) is dense in S1(E),then under some condition,every surjective isometry V0 from S1(E) onto S1(C(Ω)) can be extended to be a real linearly isometric map V of E onto C(Ω).From this resultwe also obtain some corollaries. This is the first time we study this problem on different typical spaces,and the method of proof is also very different too.