scholarly journals Ordinary, absolute and strong summability and matrix transformations

Filomat ◽  
2003 ◽  
pp. 59-78 ◽  
Author(s):  
Abdullah Jarrah ◽  
Eberhard Malkowsky
Keyword(s):  
Sufficient Conditions ◽  
Compact Operators ◽  
Strong Summability ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Serbia And Montenegro ◽  
The Past ◽  
Necessary And Sufficient ◽  
The University ◽  
Natural Way

Many important sequence spaces arise in a natural way from various concepts of summability, namely ordinary, absolute and strong summability. In the first two cases they may be considered as the domains of the matrices that define the respective methods of summability; the situation, however, is different and more complicated in the case of strong summability. Given sequence spaces X and Y, we find necessary and sufficient conditions for the entries of a matrix to map X into Y, and characterize the subclass of those matrices that are compact operators. This paper gives a survey of recent research in the field of matrix transformations at the University of Nis Serbia and Montenegro, in the past four years.

scholarly journals A note on Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations

1995 ◽  
Vol 18 (4) ◽  
pp. 681-688 ◽  
Author(s):  
B. Choudhary ◽  
S. K. Mishra
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Necessary And Sufficient

In this paper we define the sequence spacesSℓ∞(p),Sc(p)andSc0(p)and determine the Köthe-Toeplitz duals ofSℓ∞(p). We also obtain necessary and sufficient conditions for a matrixAto mapSℓ∞(p)toℓ∞and investigate some related problems.

scholarly journals Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Linear Operator ◽  
Sufficient Conditions ◽  
Sequence Spaces ◽  
General Linear ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
Complex Field ◽  
Infinite Matrices ◽  
The Matrix ◽  
Necessary And Sufficient

In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the fieldC*and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets overC*to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

scholarly journals On the generalized Riesz B-difference sequence spaces

Filomat ◽  
2010 ◽  
Vol 24 (4) ◽  
pp. 35-52 ◽  
Author(s):  
Metin Başarir
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Difference Sequence ◽  
Topological Properties ◽  
Linear Spaces ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Difference Sequence Spaces

In this paper, we define the new generalized Riesz B-difference sequence spaces rq? (p, B), rqc (p, B), rq0 (p, B) and rq (p, B) which consist of the sequences whose Rq B-transforms are in the linear spaces l?(p), c (p), c0(p) and l(p), respectively, introduced by I.J. Maddox[8],[9]. We give some topological properties and compute the ?-, ?- and ?-duals of these spaces. Also we determine the necessary and sufficient conditions on the matrix transformations from these spaces into l? and c.

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

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.

scholarly journals Matrix transformations of starshaped sequences

2001 ◽  
Vol 28 (4) ◽  
pp. 189-200
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Necessary And Sufficient

We deal with matrix transformations preserving the starshape of sequences. The main result gives the necessary and sufficient conditions for a lower triangular matrixAto preserve the starshape of sequences. Also, we discuss the nature of the mappings of starshaped sequences by some classical matrices.

scholarly journals The almost Dunford–Pettis property of semi-compact operators

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
Belmesnaoui Aqzzouz ◽  
Aziz Elbour
Keyword(s):  
Compact Operator ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractWe establish necessary and sufficient conditions for which each positive semi-compact operator (resp. the second power of a positive semi-compact operator) is almost Dunford–Pettis (resp. Dunford–Pettis).

scholarly journals On Certain Sequence Spaces

1981 ◽  
Vol 24 (2) ◽  
pp. 169-176 ◽  
Author(s):  
H. Kizmaz
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Necessary And Sufficient

AbstractIn this paper define the spaces l∞(Δ), c(Δ), and c0(Δ), where for instance l∞(Δ) = {x=(xk):supk |xk -xk + l|< ∞}, and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l∞(Δ) or c(Δ) into l∞ or c, and investigate related questions.

Masked factorable matrices

2002 ◽  
Vol 132 (1) ◽  
pp. 245-254
Author(s):  
Gord Sinnamon
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Necessary And Sufficient ◽  
Factorable Matrices

The class of masked factorable matrices is introduced and simple necessary and sufficient conditions are given for matrices in the class to represent bounded transformations between Lebesgue sequence spaces.

The Supremum of a Family of Additive Functions

1952 ◽  
Vol 4 ◽  
pp. 463-479 ◽  
Author(s):  
Israel Halperin
Keyword(s):  
Vector Space ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Boolean Ring ◽  
Additive Functions ◽  
Semi Group ◽  
Additive Functionals ◽  
Banach Theorem ◽  
Necessary And Sufficient ◽  
Natural Way

Any system S in which an addition is defined for some, but not necessarily all, pairs of elements can be imbedded in a natural way in a commutative semi-group G, although different elements in S need not always determine different elements in G (see §2). Theorem 2.1 gives necessary and sufficient conditions in order that a functional p(x) on S can be represented as the su prémuni of some family of additive functionals on S, and one such set of conditions is in terms of possible extensions of p(x) to G. This generalizes the case with 5 a Boolean ring treated by Lorentz [4], Lorentz imbeds the Boolean ring in a vector space and this could be done for the general S; but we prefer to imbed S in a commutative semi-group and to give a proof (see § 1) generalizing the classical Hahn-Banach theorem to the case of an arbitrary commutative semigroup.

On generalized Fibonacci difference sequence spaces and compact operators

2021 ◽  
pp. 2250116
Author(s):  
Taja Yaying ◽  
Bipan Hazarika ◽  
Mikail Et
Keyword(s):  
Fractional Order ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Sequence Spaces ◽  
Topological Properties ◽  
Band Matrix ◽  
Matrix Formula ◽  
Necessary And Sufficient ◽  
Matrix Mappings ◽  
Difference Sequence Spaces

In this paper, we introduce Fibonacci backward difference operator [Formula: see text] of fractional order [Formula: see text] by the composition of Fibonacci band matrix [Formula: see text] and difference operator [Formula: see text] of fractional order [Formula: see text] defined by [Formula: see text] and introduce sequence spaces [Formula: see text] and [Formula: see text] We present some topological properties, obtain Schauder basis and determine [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the spaces [Formula: see text] and [Formula: see text] We characterize certain classes of matrix mappings from the spaces [Formula: see text] and [Formula: see text] to any of the space [Formula: see text] [Formula: see text] [Formula: see text] or [Formula: see text] Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces [Formula: see text] and [Formula: see text]

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