Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Generalized difference sequence spaces and their dual spaces

The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785-791]. In this paper we compute pα- and N-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀, and compute β- and γ-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀.     

On some new generalized sequence spaces

In this paper we define the sequence sets ?_∞(u,Δ²,p), c(u,Δ²,p) and c₀(u,Δ²,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?_∞(u,Δ²,p),?_∞).     

On some new paranormed euler sequence spaces and Euler Core

In this paper, the sequence spaces e0^τ（u, p） and ec^τ（u, p） of non-absolute type which are the generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces e0^τ（u,p） and ec^τ（u,p） are linearly isomorphic to spaces co（p） and c（p）, respectively. Furthermore, the α-, β- and γ-duals of the spaces 0^τ（u,p） and ec^τ（u,p） have been computed and their bases have been constructed and some topological properties of these spaces have been investigated. Besides this, the class of matrices （e0^τ）（u, p） ： μ） has been characterized, where μ is one of the sequence spaces l∞, c and co and derives the other characterizations for the special cases of μ. In the last section, Euler Core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.     

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.     

Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness

The spaces and introduced by Ayd?n and Ba?ar [C. Ayd?n, F. Ba?ar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3) (2004) 677-693] can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero, summable, and bounded by the Cesàro method of order 1. Here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable, summable to zero, or bounded by the strong Cesàro method of order 1 with index p?1. We determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences.     

Entropy function spaces and interpolation

We associate to every function space, and to every entropy function E, a scale of spaces Λp,q(E) similar to the classical Lorentz spaces Lp,q. Necessary and sufficient conditions for they to be normed spaces are proved, their role in real interpolation theory is analyzed, and a number of applications to functional and interpolation properties of several variants of Lorentz spaces and entropy spaces are given.     

Explicit characterizations of orthogonality in lS(C)

In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l_S^p(C), 1?p<∞, thus obtaining generalizations of the usual characterization of orthogonality in the Hilbert spaces l_S²(C), via inner products. In this paper we make explicit these characterizations for some of the spaces l_S^p(C). We finish by presenting some remarks and open problems.     

A class of B-(p,r)-invex functions and mathematical programming

Invexity of a function is generalized. The new class of nonconvex functions, called B-(p,r)-invex functions with respect to η and b, being introduced, includes many well-known classes of generalized invex functions as its subclasses. Some properties of the introduced class of B-(p,r)-invex functions with respect to η and b are studied. Further, mathematical programming problems involving B-(p,r)-invex functions with respect to η and b are considered. The equivalence between saddle points and optima, and different type duality theorems are established for this type of optimization problems.     

The strongest intrinsic meaning of sequential-evaluation convergence

For classical Banach sequence spaces c₀(X), l_∞(X) and lp(X) (0<p<+∞) we have found the strongest intrinsical meanings of their β-duals, and two basic convergence results are established in the β-duals.     

On L-convergence of trigonometric series

In the present paper we consider the trigonometric series with (β,r)-general monotone and (β,r)-rest bounded variation coefficients. Necessary and sufficient conditions of L-convergence for such series are obtained in terms of the coefficients. Moreover, we generalize and extend the Tikhonov results [J. Math. Anal. Appl. 347 (2008) 416-427] to the class GM(β,r) or the class RBVS(β,r).     

On the m order difference sequence space of generalized weighted mean and compact operators

In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space ?(u, v, p; Δ^(m)), which consist of the sequences whose generalized weighted Δ^(m)-difference means are in the linear space ?(p) defined by I.J. Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from ?(u, v, p, Δ^(m)) to ?_∞, c and c₀. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space ?_p(u, v, Δ^(m))(1 ≤ p < ∞).     

Some topological and geometric properties of generalized Euler sequence space

In this paper, we introduce the Euler sequence space e ^r (p) of nonabsolute type and prove that the spaces e ^r (p) and l(p) are linearly isomorphic. Besides this, we compute the α-, β- and γ-duals of the space e ^r (p). The results proved herein are analogous to those in [ALTAY, B.—BASŠAR, F.: On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26 (2002), 701–715] for the Riesz sequence space r ^q (p). Finally, we define a modular on the Euler sequence space e ^r (p) and consider it equipped with the Luxemburg norm. We give some relationships between the modular and Luxemburg norm on this space and show that the space e ^r (p) has property (H) but it is not rotund (R).     

Subnormality of Bergman-like weighted shifts

For a,b,c,d?0 with ad−bc>0, we consider the unilateral weighted shift S(a,b,c,d) with weights . Using Schur product techniques, we prove that S(a,b,c,d) is always subnormal; more generally, we establish that for every p?1, all p-subshifts of S(a,b,c,d) are subnormal. As a consequence, we show that all Bergman-like weighted shifts are subnormal.     

Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces

In this paper, the necessary and sufficient conditions are found for the boundedness of the rough B-fractional integral operators from the Lorentz spaces Lp,s,γ to Lq,r,γ, 1<p<q<∞, 1?r?s?∞, and from L1,r,γ to Lq,∞,γ≡WLq,γ, 1<q<∞, 1?r?∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.     

A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces

Let s∈R, τ∈[0,∞), p∈(1,∞) and q∈(1,∞]. In this paper, we introduce a new class of function spaces which unify and generalize the Triebel-Lizorkin spaces with both p∈(1,∞) and p=∞ and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel-Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004) 377-422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where s∈R, p,q∈[1,∞), max{p,q}>1, , and t^′ denotes the conjugate index of t∈(1,∞); as an application of this, we further introduce certain Hardy-Hausdorff spaces and prove that the dual space of is just when p,q∈(1,∞).     

Some inequalities involving upper bounds for some matrix operators I

In this paper we consider the problem of finding upper bounds of certain matrix operators such as Hausdorff, Nörlund matrix, weighted mean and summability on sequence spaces l _p(w) and Lorentz sequence spaces d(w, p), which was recently considered in [9] and [10] and similarly to [14] by Josip Pecaric, Ivan Peric and Rajko Roki. Also, this study is an extension of some works by G. Bennett on l _p spaces, see [1] and [2].     

Embeddings and frames of function spaces

For every Tychonoff space X we denote by Cp(X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of RX. A set P is a frame for the space Cp(X) if Cp(X)⊂P⊂RX. We prove that if Cp(X) embeds in a σ-compact space of countable tightness then X is countable. This shows that it is natural to study when Cp(X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space Cp(X) has a Lindelöf frame of countable tightness then t(X)?ω. We give some generalizations of this result for the case of frames as well as for embeddings of Cp(X) in arbitrary spaces.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

Bandwidth theorem for random graphs

A graph G is said to have bandwidth at most b, if there exists a labeling of the vertices by 1,2,…,n, so that |i−j|?b whenever {i,j} is an edge of G. Recently, Böttcher, Schacht, and Taraz verified a conjecture of Bollobás and Komlós which says that for every positive r, Δ, γ, there exists β such that if H is an n-vertex r-chromatic graph with maximum degree at most Δ which has bandwidth at most βn, then any graph G on n vertices with minimum degree at least (1−1/r+γ)n contains a copy of H for large enough n. In this paper, we extend this theorem to dense random graphs. For bipartite H, this answers an open question of Böttcher, Kohayakawa, and Taraz. It appears that for non-bipartite H the direct extension is not possible, and one needs in addition that some vertices of H have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed r-chromatic graph H₀ which one can find in a spanning subgraph of G(n,p) with minimum degree (1−1/r+γ)np.