On Domain of Nörlund Sequences

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α, β, γduals, and characterized their matrix transformations on this space and into this space.

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The Hahn Sequence Space Defined by the Cesáro Mean

Abstract and Applied Analysis ◽

10.1155/2013/817659 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Murat Kirişci

Keyword(s):

Banach Space ◽

Sequence Space ◽

Matrix Transformations ◽

Space Theory ◽

Topological Properties ◽

Cesàro Mean ◽

First Order ◽

Inclusion Relations ◽

The Matrix ◽

Cesaro Mean

The -space of all sequences is given as such that converges and is a null sequence which is called the Hahn sequence space and is denoted by . Hahn (1922) defined the space and gave some general properties. G. Goes and S. Goes (1970) studied the functional analytic properties of this space. The study of Hahn sequence space was initiated by Chandrasekhara Rao (1990) with certain specific purpose in the Banach space theory. In this paper, the matrix domain of the Hahn sequence space determined by the Cesáro mean first order, denoted by , is obtained, and some inclusion relations and some topological properties of this space are investigated. Also dual spaces of this space are computed and, matrix transformations are characterized.

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On matrix transformations concerning the Nakano vector-valued sequence space

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201010730 ◽

2001 ◽

Vol 26 (11) ◽

pp. 671-678

Author(s):

Suthep Suantai

Keyword(s):

Sequence Space ◽

Sequence Spaces ◽

Matrix Transformations ◽

Real Numbers ◽

Positive Real ◽

The Matrix ◽

Vector Valued ◽

Bounded Sequences

We give the matrix characterizations from Nakano vector-valued sequence spaceℓ(X,p)andFr(X,p)into the sequence spacesEr,ℓ∞,ℓ¯∞(q),bs, andcs, wherep=(pk)andq=(qk)are bounded sequences of positive real numbers such thatPk>1for allk∈ℕandr≥0.

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On Some Properties of New Paranormed Sequence Space of Nonabsolute Type

Abstract and Applied Analysis ◽

10.1155/2012/921613 ◽

2012 ◽

Vol 2012 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Vatan Karakaya ◽

Necip Şimşek

Keyword(s):

Sequence Space ◽

Sequence Spaces ◽

The Other ◽

Matrix Transformations ◽

Topological Properties ◽

Paranormed Sequence Space ◽

Inclusion Relations

We introduce some new generalized sequence space related to the space . Furthermore we investigate some topological properties as the completeness, the isomorphism, and also we give some inclusion relations between this sequence space and some of the other sequence spaces. In addition, we compute -, -, and -duals of this space and characterize certain matrix transformations on this sequence space.

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Matrix transformations involving certain Banach space valued sequence spaces

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.31.2000.400 ◽

2000 ◽

Vol 31 (2) ◽

pp. 85-100

Author(s):

J. K. Srivastava ◽

B. K. Srivastava

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Linear Operators ◽

Sequence Spaces ◽

Matrix Transformations ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Matrix Classes ◽

The Matrix ◽

Matrix Class

In this paper for Banach spaces $X$ and $Y$ we characterize matrix classes $ (\Gamma (X,\lambda)$, $ l_\infty(Y,\mu))$, $ (\Gamma(X,\lambda),C(Y,\mu))$, $ (\Gamma(X,\lambda)$, $ c_0(Y,\mu))$, $ (\Gamma(X,\lambda)$, $ \Gamma^*(Y,\mu))$, $ (l_1(X,\lambda)$, $ \Gamma(Y,\mu))$ and $ (c_0(X,\lambda)$, $ c_0(Y,\mu))$ of bounded linear operators involving $ X$- and $ Y$-valued sequence spaces. Further as an application of the matrix class $ (c_0(X,\lambda)$, $ c_0(Y,\mu))$ we investigate the Banach space $ B(c_0(X,\lambda)$, $ c_0(Y,\mu))$ of all bounded linear mappings of $ c_0(x,\lambda)$ to $ c_0(Y,\mu)$.

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Approximation numbers of matrix transformations and inclusion maps

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.42.2011.924 ◽

2011 ◽

Vol 42 (2) ◽

pp. 193-203

Author(s):

M. Gupta ◽

L. R. Acharya

Keyword(s):

Banach Space ◽

Sequence Space ◽

Sequence Spaces ◽

Matrix Transformations ◽

Diagonal Operator ◽

Approximation Numbers ◽

Vector Valued

In this paper we establish relationships of the approximation numbers of matrix transformations acting between the vector-valued sequence spaces spaces of the type $\lambda(X)$ defined corresponding to a scalar-valued sequence space $\lambda$ and a Banach space $(X,\|.\|)$ as $$\lambda(X)=\{\overline x=\{x_i\}: x_i\in X, \forall~i\in \mathbb{N},~\{\|x_i\|_X\}\in \lambda\};$$ with those of their component operators. This study leads to a characterization of a diagonal operator to be approximable. Further, we compute the approximation numbers of inclusion maps acting between $\ell^p(X)$ spaces for $1\leq p\leq \infty$.

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On some generalized difference sequences with respect to a modulus function

Filomat ◽

10.2298/fil0317023e ◽

2003 ◽

pp. 23-33 ◽

Cited By ~ 9

Author(s):

Mikail Et ◽

Yavuz Altin ◽

Hifsi Altinok

Keyword(s):

Banach Space ◽

Sequence Spaces ◽

Modulus Function ◽

Difference Sequence ◽

Difference Sequences ◽

Inclusion Relations ◽

Difference Sequence Spaces

The idea of difference sequence spaces was intro- duced by Kizmaz [9] and generalized by Et and Colak [6]. In this paper we introduce the sequence spaces [V, ?, f, p]0 (?r, E), [V, ?, f, p]1 (?r, E), [V, ?, f, p]? (?r, E) S? (?r, E) and S?0 (?r, E) where E is any Banach space, examine them and give various properties and inclusion relations on these spaces. We also show that the space S? (?r, E) may be represented as a [V, ?, f, p]1 (?r, E)space.

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Domain of the Double Sequential Band Matrix in the Sequence Space

Abstract and Applied Analysis ◽

10.1155/2013/949282 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Havva Nergiz ◽

Feyzi Başar

Keyword(s):

Sequence Space ◽

Sequence Spaces ◽

Schauder Basis ◽

Matrix Transformations ◽

Difference Matrix ◽

Band Matrix ◽

Alpha Beta

The sequence space was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix in the sequence space has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space of nonabsolute type has been studied which is the domain of the generalized difference matrix in the sequence space . Furthermore, the alpha-, beta-, and gamma-duals of the space have been determined, and the Schauder basis has been given. The classes of matrix transformations from the space to the spaces ,candc0have been characterized. Additionally, the characterizations of some other matrix transformations from the space to the Euler, Riesz, difference, and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to conclusion.

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On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

Mathematica Slovaca ◽

10.1515/ms-2021-0059 ◽

2021 ◽

Vol 71 (6) ◽

pp. 1375-1400

Author(s):

Feyzi Başar ◽

Hadi Roopaei

Keyword(s):

Lower Bound ◽

Sequence Space ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Matrix Transformations ◽

Infinite Matrix ◽

The Matrix ◽

Alpha Beta ◽

Necessary And Sufficient ◽

Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

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