The generalized multiplier space and its Köthe-Toeplitz and null duals (original) (raw)
2020
We will start with the set M(X,Y)M(X,Y)M(X,Y), multiplier space, defined by: \[ M(X,Y)=\{a=(a_k)\in \omega \mid ax\in Y \mbox{, for all }x\in X\} \] where omega\omegaomega denotes the space of all complex-valued sequences and XXX and YYY are sequence spaces. Specially, putting Y=csY=csY=cs, where cscscs is the set of convergent series, the multiplier space becomes the beta\betabeta-dual of XXX. We will present some generalized results related to XbetaX^{\beta}Xbeta and extend some of existing. Finally, we will illustrate these generalizations with some examples and applications.
The αAB-, βAB-, γab- and NAB-duals for sequence spaces
Filomat, 2017
Let A = (an,k) and B = (bn,k) be two infinite matrices with real entries. The main purpose of this paper is to generalize the multiplier space for introducing the concepts of ?AB-, ?AB-, ?AB-duals and NAB-duals. Moreover, these duals are investigated for the sequence spaces X and X(A), where X ? {c0, c, lp} for 1 ? p ? ?. The other purpose of the present study is to introduce the sequence spaces X(A,?) = {x=(xk): (?x?k=1 an,kXk - ?x?k=1 an-1,kXk)? n=1 ? X}, where X ? {l1,c,c0}, and computing the NAB-(or Null) duals and ?AB-duals for these spaces.
Generalized Köthe-Toeplitz Duals of Some Vector-Valued Sequence Spaces
International Journal of Analysis, 2013
We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces.
Journal of Inequalities and Applications, 2013
The purpose of this paper is to introduce new sequence spaces associated with a multiplier sequence by using an infinite matrix, an Orlicz function and a generalized B-difference operator on a real n-normed space. Some topological properties of these spaces are examined. We also define a new concept, which will be called (B μ ) n -statistical A-convergence, and establish some inclusion connections between the sequence space W(A, B μ , p, ·, . . . , · ) and the set of all (B μ ) n -statistically A-convergent sequences. MSC: Primary 40A05; secondary 40B50; 46A19; 46A45
On Certain Vector Valued Multiplier Spaces and Series of Operators
2019
By L(X,Y ), we denote the space of all continuous linear operators between the normed spaces X and Y . In [15], Swartz introduced the (bounded) multiplier space for the series ∑ Tj as: M∞( ∑ Tj) = {x = (xj) ∈ `∞(X)| ∑ j Tjxj converges}, where (Tj) ⊆ L(X,Y ). Recently in [6], Altay and Kama defined the vector valued multiplier space M∞ C (T ) of Cesàro convergence by using Cesàro summability method as follow: M∞ C (T ) = {x = (xk) ∈ `∞(X)| ∑ k Tkxk is Cesàro convergent}. In this paper, we introduce the vector valued multiplier spaces SΛ(T ) and SwΛ(T ) by means of Λ− convergence and a sequence of continuous linear operators and study a series of some properties of these spaces.
KÖTHE TÖEPLITZ DUALS OF CERTAIN BICOMPLEX SEQUENCE SPACES
TJPRC, 2014
In this paper, we have investigated certain bicomplex duals of the class of bicomplex sequences defined by Srivastava & Srivastava [S2] and of the subclasses defined by Nigam [N1].Two types of duals namelya - dual and b - dual, given by Köthe & Töeplitz [K2], have been defined for bicomplex sequence spaces. Relations between these duals have been studied. Other two types of duals have also been defined. The inclusions have been shown to be proper by means of counter examples.
Convolution in dual Cesàro sequence spaces
Journal of Mathematical Analysis and Applications
We investigate convolution operators in the sequence spaces d p , for 1 ≤ p < ∞. These spaces, for p > 1, arise as dual spaces of the Cesàro sequence spaces ces p thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve d p into d p. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of ℓ p .
On Köthe-Toeplitz duals of generalized difference sequence spaces
Bull. Malaysian Math. Sci. Soc, 2000
Abstract. In this paper, we define the sequence spaces)(,)(cmvmv" and,)(,)(N m co mv and give some topological properties, inclusion relations of these sequence spaces, compute their continuous and Köthe-Toeplitz duals. The results of this paper, in a particular case, ...
The n-dual space of the space of p-summable sequences
Mathematica Bohemica
In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an n-normed space, we are interested in bounded multilinear n-functionals and n-dual spaces. The concept of bounded multilinear n-functionals on an n-normed space was initially introduced by A. G. White jun. [Math. Nachr. 42, 43–60 (1969; Zbl 0185.20003)] and studied further by H. Batkunde, H. Gunawan and Y. E. P. Pangalela [“Bounded linear functionals on the n-normed space of p-summable sequences”, Acta Univ. M. Belii, Ser. Math. 21, 66–75 (2013), http://actamath.savbb.sk/pdf/aumb2107.pdf\] and S. M. Gozali et al. [Ann. Funct. Anal. AFA 1, No. 1, 72–79 (2010; Zbl 1208.46006)]. In this paper, we revisit the definition of bounded multilinear n-functionals, introduce the concept of n-dual spaces, and then determine the n-dual spaces of ℓ p spaces when these spaces are not only equipped with the usual norm, but also with some n-norms.
Cesàro Orlicz Sequence Spaces and Their Köthe-Toeplitz Duals
2018
The present paper focus on introducing certain classes of Cesàro Orlicz sequences over n-normed spaces. We study some topological and algebraic properties of these spaces. Further, we examine relevant relations among the classes of these sequences. We show that these spaces are made n-BK-spaces under certain conditions. Finally, we compute the Köthe-Toeplitz duals of these spaces.
Generalized difference sequence spaces and their dual spaces
Journal of Mathematical Analysis and Applications, 2004
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 ∆ m v (X) for X = ∞ , c and c 0 , and compute β-and γ-duals of the sequence spaces ∆ m v (X) for X = ∞ , c and c 0 .
Multipliers on Generalized Mixed Norm Sequence Spaces
Abstract and Applied Analysis, 2014
Given1≤p,q≤∞and sequences of integers(nk)kand(nk′)ksuch thatnk≤nk′≤nk+1, the generalized mixed norm spaceℓℐ(p,q)is defined as those sequences(aj)jsuch that((∑j∈Ik|aj|p)1/p)k∈ℓqwhereIk={j∈ℕ0 s.t. nk≤j
I-convergent sequence spaces associated with multiplier sequences
Mathematical Inequalities & Applications, 2008
In this article we introduce the sequence spaces c I (Λ) , c I 0 (Λ) , m I (Λ) and m I 0 (Λ) associated with the multiplier sequence Λ = (λ k) of non-zero scalars. We study the different algebraic and topological properties of these sequence spaces like solidness, symmetricity, sequence algebra, convergence free etc. Also we characterize the multiplier problem and obtain some inclusion relation involving these sequence spaces.
On Multipliers in Sequence Spaces and a Theorem of Bosanquet
Analysis, 1989
For sequence spaces £ and F, a sequence u is an (E F) multiplier when e E always implies (u^x^) e F. When £ and F are matrix fields, u is often called a summability-factor sequence, and there are many classical results in such cases. We give some general results on multipliers, depending on basis arguments, and illustrate these with a discussion of a well-known theorem of Bosanquet on Cesàro summability factors with order conditions.
On the Classical Paranormed Sequence Spaces and Related Duals over the Non-Newtonian Complex Field
Journal of Function Spaces, 2015
The studies on sequence spaces were extended by using the notion of associated multiplier sequences. A multiplier sequence can be used to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. Sometimes the associated multiplier sequence delays the rate of convergence of a sequence. In the present paper, the classical paranormed sequence spaces have been introduced and proved that the spaces are⋆-complete. By using the notion of multiplier sequence, theα-,β-, andγ-duals of certain paranormed spaces have been computed and their basis has been constructed.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2018
The Fréchet sequence spaces ces(p+) are very different to the Fréchet sequence spaces p+ , 1 ≤ p < ∞, that generate them, [3]. The aim of this paper is to investigate various properties (eg. continuity, compactness, mean ergodicity) of certain linear operators acting in and between the spaces ces(p+), such as the Cesàro operator, inclusion operators and multiplier operators. Determination of the spectra of such classical operators is an important feature. It turns out that both the space of multiplier operators M(ces(p+)) and its subspace M c (ces(p+)) consisting of the compact multiplier operators are independent of p. Moreover, M c (ces(p+)) can be topologized so that it is the strong dual of the Fréchet-Schwartz space ces(1+) and (M c (ces(p+)) β ces(1+) is a proper subspace of the Köthe echelon Fréchet space M(ces(p+)) = λ ∞ (A), 1 ≤ p < ∞, for a suitable matrix A.