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AK(V)-Property of Double Series Spaces
(Springernature, 2021) Yesilkayagil, Medine; Basar, Feyzi
In the present paper, we show that V-convergent double series spaces CSV and the series space BV of double sequences of bounded variation are BDK-spaces and investigate their AK (V)-space property, where V is an element of {p, bp, r}.
Banach Spaces and Inequalities Associated with New Generalization of Cesaro Matrix
(Springer, 2023) Basar, Feyzi; Roopaei, Hadi
Let the triangle matrix A(ru) be a generalization of the Cesaro matrix and U & ISIN; {c(0), c, l(& INFIN;)}. In this study, we essentially deal with the space U(A(ru)) defined by the domain of A(ru) in the space U and give the bases, and determine the Kothe-Toeplitz, generalized Kothe-Toeplitz and bounded-duals of the space U (A(ru)). We characterize the classes (l(& INFIN;) (A(ru)):l(& INFIN;)), (l(& INFIN;)(A(ru)): c), (c(A(ru)): c), and (U: V(A(ru))) of infinite matrices, where V denotes any given sequence space. Additionally, we also present a Steinhaus type theorem. As an another result of this study, we investigate the l(p)-norm of the matrix A(ru) and as a result obtaining a generalized version of Hardy's inequality, and some inclusion relations. Moreover, we compute the norm of well-known operators on the matrix domain l(p) (A(ru)).
Characterizations of Unconditionally Convergent andWeakly Unconditionally Cauchy Series via wRp -Summability, Orlicz-Pettis Type Theorems and Compact Summing Operator
(Univ Nis, Fac Sci Math, 2022) Karakus, Mahmut; Basar, Feyzi
In the present paper, we give a new characterization of unconditional convergent series and give some new versions of the Orlicz-Pettis theorem via FQ s-family and a natural family F with the separation property S1 through wRp -summability which may be considered as a generalization of the well-known strong p-Ces`aro summability. Among other results, we obtain a new (weak) compactness criteria for the summing operator.
Domain of Riesz mean in some spaces of double sequences
(Elsevier Science Bv, 2018) Yesilkayagil, Medine; Basar, Feyzi
In this study, we define the double sequence spaces (M-u)(Rqt), (Cp)(Rqt), (C-bp)(Rqt) and (C-r)(Rqt) as the domain of four dimensional Riesz mean R-qt in the spaces M-u, C-p, C-bp and C-r, respectively. Then, we examine some topological properties of those sequence spaces and we characterize the RH-regularity of the Riesz mean R-qt. Taking v is an element of{p, bp, r}, we determine the beta(v)-duals of the spaces (C-v)(Rqt) and we characterize the classes ((C-r)(Rqt) : C-v), (mu : (C-v)(Rqt)) and ((C-v)(Rqt) : C-f) of four-dimensional matrix transformations, where mu and C-f denote any given double sequence space and the space of almost convergent double sequences, respectively. (C) 2018 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).
DOMAIN OF THE CESARO MEAN OF ORDER ? IN MADDOX'S SPACE l(p)
(Publications L Institut Mathematique Matematicki, 2023) Savasci, Medine Yesilkayagil; Basar, Feyzi
The sequence space l(p) was defined by I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2), 18 (1967), 345-355. Here, we introduce the paranormed Cesaro sequence space l(C-alpha, p) of order alpha, of non-absolute type as the domain of Cesaro mean C-alpha of order alpha and prove that the spaces l (C-alpha, p) and l(p) are linearly paranorm isomorphic. Besides this, we compute the alpha-, beta- and gamma-duals of the space l(C-alpha, p) and construct the basis of the space l(C-alpha, p) together with the characterization of the classes of matrix transformations from the space l(C-alpha, p) into the spaces l(infinity) of bounded sequences and f of almost convergent sequences, and any given sequence space Y, and from a given sequence space Y into the sequence space l(C-alpha, p). Finally, we emphasize on some geometric properties of the space l(C-alpha, p).
A generalization of almost convergence, completeness of some normed spaces with wuC series and a version of Orlicz-Pettis theorem
(Springer-Verlag Italia Srl, 2019) Karakus, Mahmut; Basar, Feyzi
In this study, we give a slight generalization of almost convergence and introduce some new multiplier spaces associated to a series k xk in a normed space X by means of this new summability method. We also obtain some characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in X and X *, respectively. Finally, we give a version of the Orlicz-Pettis theorem, as an application of this new method and an unconditionally convergent series k xk in a normed space X.
The Hahn sequence space generated by the Cesaro mean of order m
(Springer Birkhauser, 2024) Savasci, Medine Yesilkayagil; Basar, Feyzi
Hahn (Math Phys 32:3-88, 1922) defined the sequence space h. The main purpose of this study is to introduce the new Hahn sequence space h(C-m) as the domain of Cesaro mean of order m and give some topological properties of the space h(Cm). Moreover, we determine the alpha-, beta-and gamma-duals of the space h(C-m) and characterize the classes (i1 : h), (h : 4), (h(C-m) : V-1) and (V-2 : h(Cm)) of matrix transformations, where 1 < p < infinity, V-1 is an element of {l(infinity), c, c(0), l(p)} and V-2 is any given sequence space. Finally, we compute the norm of the operators belonging to B(e(1), h(C-m)) and determine the Hausdorff measure of noncompactness of the operators in B(i(1), h(C-m)).
Matrix Transformation and Compactness on q-Catalan Sequence Spaces
(Taylor & Francis Inc, 2024) Yaying, Taja; Basar, Feyzi
This article intends to develop q-Catalan sequence spaces l(p)(C(q)) and l(infinity)(C(q)) due to q-Catalan matrix C(q) in lp and l infinity, respectively. Apart from obtaining some basic topological properties and Schauder basis, we compute alpha-, beta-, and gamma-duals of the spaces l(p)(C(q)) and l(infinity)(C(q)). We state and prove a theorem that characterize certain matrix classes (X, Y), where X is either of the space l(p)(C(q)) or l(infinity)(C(q)) and Y is an element of{l(infinity),c(0),c,l(1)}. The final section is devoted to determination of certain conditions by which a matrix operator becomes compact.
Matrix transformations on some sequence spaces related to strong Cesaro summability and boundedness
(Elsevier Science Inc, 2009) Altay, Bilal; Basar, Feyzi; Malkowsky, Eberhard
The spaces a(0)(r)(Delta), a(c)(r)(Delta) and a(infinity)(r)(Delta) introduced by Aydin and Basar [ C. Aydin, F. Basar, 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 Cesaro method of order 1. Here we de. ne 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 Cesaro method of order 1 with index p >= 1. We determine the beta-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences. (c) 2009 Elsevier Inc. All rights reserved.
Matrix transformations on the matrix domains of triangles in the spaces of strongly C1-summable and bounded sequences
(Kossuth Lajos Tudomanyegyetem, 2008) Basar, Feyzi; Malkowsky, Eberhard; Altay, Bilal
Let w(0)(p), w(p) and w(infinity)(p) be the sets of sequences that are strongly summable to zero, summable and bounded of index p >= 1 by the Cesaro method of order 1, which were introduced by Maddox [I. J. MADDOX, On Kuttner's theorem, J. London Math. Soc. 43 (1968), 285-290]. We study the matrix domains w(0)(p)(T) = (W-0(p))(T), w(p)(T) = (W-p)T and w(infinity)(p) (T) = (W-infinity(p))T of arbitrary triangles T in w(0)(p),w(p) and w(infinity)(p), determine their beta-duals, and characterize matrix transformations on them into the spaces c(0), c and l(infinity).
A NOTE ON SOME TOPOLOGICAL PROPERTIES OF KOTHE SPACE ?(P)
(Publications L Institut Mathematique Matematicki, 2019) Yesilkayagil, Medine; Basar, Feyzi
We emphasize some topological properties of the Kothe space lambda(P) determined by a Kothe set P.
ON SOME CLASSICAL PROPERTIES OF NORMED SPACES VIAGENERALIZED VECTOR VALUED ALMOST CONVERGENCE
(Walter De Gruyter Gmbh, 2022) Karakus, Mahmut; Basar, Feyzi
Recently, the authors interested some new problems on multiplier spaces of Lorentz' almost convergence and f(lambda)-convergence as a generalization of almost convergence. f(lambda)-convergence is firstly introduced by Karakus and Basar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space X and its continuous dual X*. In the present paper, we deal with f(lambda)-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces.
ON SOME LAMBDA-PASCAL SEQUENCE SPACES AND COMPACT OPERATORS
(Rocky Mt Math Consortium, 2022) Yaying, Taja; Basar, Feyzi
We introduce Lambda-Pascal sequence spaces l(q) (G), c0(G), c(G) and l8(G) generated by the matrix G which is obtained by the product of Pascal matrix and 3-matrix. It is proved that the Lambda-Pascal sequence spaces l (q) ( G), c(0)(G), c(G) and l(infinity)(G) are BK-spaces and linearly isomorphic to l (q), c(0), c and l(infinity), respectively. We construct Schauder bases and obtain alpha-, ss- and gamma-duals of the new spaces. We state and prove characterization theorems related to matrix transformation from the space l (q) (G) to the spaces l(infinity), c and c(0). Finally, we determine necessary and sufficient conditions for a matrix operator to be compact from the space c(0)(G) to any one of the spaces l(infinity), c, c(0) or l(1).
ON SOME NEW SEQUENCE SPACES DEFINED BY q-PASCAL MATRIX
(Ivane Javakhishvili Tbilisi State Univ, 2022) Yaying, Taja; Hazarika, Bipan; Basar, Feyzi
In this study, we construct the q-analog P(q) of Pascal matrix and study the sequence spaces c(P(q)) and c(0)(P(q)) defined as the domain of q-Pascal matrix P(q) in the spaces c and c(0), respectively. We investigate certain topological properties, determine Schauder bases and compute Kothe duals of the spaces c(0)(P(q)) and c(P(q)). We state and prove the theorems characterizing the classes of matrix mappings from the space c(P(q)) to the spaces l(infinity) of bounded sequences and f of almost convergent sequences. Additionally, we also derive the characterizations of some classes of infinite matrices as a direct consequence of the results about the classes (c(P (q)), l(infinity)) and (c(P(q)), f)). Finally, we obtain the necessary and sufficient conditions for a matrix operator to be compact from the space c(0)(P (q)) to anyone of the spaces l(infinity), c, c(0), l(1), cs(0), cs, bs.
On Some Spaces Isomorphic to the Space of Absolutely q-summable Double Sequences
(Kyungpook Natl Univ, Dept Mathematics, 2018) Capan, Husamettin; Basar, Feyzi
Let 0 < q < infinity. In this study, we introduce the spaces BVq and LSq of q-bounded variation double sequences and q-summable double series as the domain of four-dimensional backward difference matrix Delta and summation matrix S in the space L-q of absolutely q-summable double sequences, respectively. Also, we determine their alpha- and beta-duals and give the characterizations of some classes of four-dimensional matrix transformations in the case 0 < q <= 1.
On the difference spaces of almost convergent and strongly almost convergent double sequences
(Springer, 2019) Capan, Husamettin; Basar, Feyzi
In this paper, we study the difference spaces F(), F0(), [F]() and [F]0() of double sequences obtained as the domain of four-dimensional backward difference matrix in the spaces F, F0, [F] and [F]0 of almost convergent, almost null, strongly almost convergent and strongly almost null double sequences; respectively. We examine general topological properties of those spaces and give some inclusion theorems. Furthermore, we deal with their dual spaces.
On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
(Walter De Gruyter Gmbh, 2021) Basar, Feyzi; Roopaei, Hadi
Let F denote the factorable matrix and X is an element of {l(p), c(0), c, l(infinity)}. 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(infinity)), (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 Cesaro 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.
On the Gamma Spaces Including the Spaces of Absolutely p-Summable, Null, Convergent and Bounded Sequences
(Taylor & Francis Inc, 2022) Roopaei, Hadi; Basar, Feyzi
In this paper, we investigate some properties of the domains l(p) (Gamma(n)), c(0) (Gamma(n)), c(Gamma(n)) and l(infinity)(Gamma(n)) of the Gamma matrix of order n in the classical spaces l(p), C-0, c and l(infinity) of absolutely p-summable, null, convergent and bounded sequences, respectively, and compute the alpha-, beta- and gamma-duals of these spaces. We characterize the classes of infinite matrices from the space l(p) (Gamma(n)) to the spaces l(infinity) and f, and from a normed sequence space to the gamma sequence spaces l(p)(Gamma(n)), c(0) (Gamma(n)), c(Gamma(n)) and l(infinity)(Gamma(n)). Moreover, we introduce the necessary and sufficient conditions for factorizing an operator based on the weighted mean matrices and derive the factorizations for the Cesaro and Hilbert matrices based on the Gamma matrix. Finally, we emphasize on the lower bound of operators from l(p) into l(p) (Gamma(n)), from l(p)(Gamma(n)) into l(p), from l(p) (Gamma(n)) into itself and from l(p) into itself.
On the Paranormed Space L(t) of Double Sequences
(Univ Nis, Fac Sci Math, 2018) Capan, Husamettin; Basar, Feyzi
In this paper, we introduce the paranormed sequence space L(t) which is the generalization of the space L q of all absolutely q summable double sequences. We examine some topological properties of the space L(t) and determine its alpha-, beta -and gamma-duals. Finally, we characterize some classes of four-dimensional matrix transformations from the space L(t) into some spaces of double sequences.
On the Paranormed Space of Bounded Variation Double Sequences
(Springernature, 2020) Yesilkayagil, Medine; Basar, Feyzi
In this study, as the domain of four-dimensional backward difference matrix in the space Lu(t), we introduce the complete paranormed space BV(t) of bounded variation double sequences and examine some properties of that space. Also, we determine the gamma-dual and beta(theta)-dual of the space BV(t). Finally, we characterize the classes (BV(t):Mu), (BV(t):C theta) and (Lu(t):mu) with mu is an element of{BS,CS theta,Mu(Delta),C theta(Delta)}, where Mu(Delta) and C theta(Delta) denote the spaces of all double sequences whose Delta-transforms are in the spaces Mu and C theta, respectively.

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