Browsing by Author "Karakus, Mahmut"

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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.
Spaces of Multiplier Σ-Convergent Vector Valued Sequences and Uniform Σ-Summability
(Amer inst Mathematical Sciences-aims, 2025) Karakus, Mahmut
This study focuses on the development of novel vector-valued sequence spaces whose elements are characterized by constructing (weakly) multiplier sigma-convergent series. To achieve this, the concept of invariant means is rigorously examined and utilized as a foundational tool. These newly defined spaces are proven to possess the structure of Banach spaces when equipped with their natural sup norm, thus ensuring their completeness. In addition to establishing the Banach space properties, this study delves into the inclusion relationships between these new sequence spaces and classical multiplier spaces, specifically BMC(B) and CMC(B), where B denotes an arbitrary Banach space. By employing the sigma-convergence method, this study also culminates in a result analogous to the celebrated Hahn-Schur theorem, which traditionally establishes a connection between the weak convergence and the uniform convergence of unconditionally convergent series.
On Generalized Quasi-Convex Bounded Sequences
(Amer inst Physics, 2016) Karakus, Mahmut
The space of all sequences a = (a(k)) for which parallel to a parallel to(q) = Sigma(k)k vertical bar Delta(2) a(k)vertical bar + sup(k) vertical bar a(k)vertical bar < infinity is denoted by q. Here, Delta a(k) = a(k) - a(k+1) and Delta(m)a(k) = Delta(Delta(m-1) a(k)) = Delta(m-1) a(k) - Delta(m-1) a(k+1) with Delta(0)a(k) = a(k), m >= 1. If a = (a(k)) is an element of q then k Delta a(k) -> 0 (k -> infinity) and q subset of by, the space of all sequences of bounded-variation, since Sigma vertical bar Delta a(k)vertical bar <= Sigma(k)k vertical bar Delta(2) a(k)vertical bar. In this study, we give a generalization of quasi-convex bounded sequences.
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.
Operator Valued Series, Almost Summability of Vector Valued Multipliers and (Weak) Compactness of Summing Operator
(Academic Press inc Elsevier Science, 2020) Karakus, Mahmut; Basar, Feyzi
In this study, we introduce the vector valued multiplier spaces M-f(infinity)(Sigma T-k(k) ) and M-wf(infinity)(Sigma(k) T-k) by means of almost summability and weak almost summability, and a series of bounded linear operators. Since these multiplier spaces are equipped with the sup norm and are subspaces of l(infinity) (X), we obtain the completeness of a normed space via the multiplier spaces which are complete for every c(0) (X)-multiplier Cauchy series. We also characterize the continuity and (weakly) compactness of the summing operator S from the multiplier spaces M-f(infinity)(Sigma T-k(k) ) or M-wf(infinity)(Sigma(k) T-k) to an arbitrary normed space Y through c(0) (X)-multiplier Cauchy and too (X)-multiplier convergent series, respectively. Finally, we show that if Sigma(k) T-k is l(infinity) (X)-multiplier Cauchy, then the multiplier spaces of almost convergence and weak almost convergence are identical. These results are more general than the corresponding consequences given by Swartz [20], and are analogues given by Altay and Kama [6]. (C) 2019 Published by Elsevier Inc.
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.
On Certain Vector Valued Multiplier Spaces and Series of Operators
(Univ Prishtines, 2019) Karakus, Mahmut
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 Sigma T-j as: M-infinity (Sigma T-j) = {x = (x(j)) is an element of l(infinity) (X)vertical bar Sigma(j)T(j)x(j) converges}, where (T-j) subset of L (X, Y). Recently in [6], Altay and Kama de fined the vector valued multiplier space M-C(infinity) (T) of Cesaro convergence by using Cesaro summability method as follow: M-C(infinity) (T) = {x = (x(k)) is an element of l(infinity) (X)vertical bar Sigma(k)T(j)x(j) converges}, In this paper, we introduce the vector valued multiplier spaces S-Lambda (T) and S-w Lambda (T) by means of Lambda- convergence and a sequence of continuous linear operators and study a series of some properties of these spaces.
Vector Valued Multiplier Spaces of Fλ-Summability, Completeness Through C0(x)-Multiplier Convergence and Continuity and Compactness of Summing Operators
(Springer-verlag Italia Srl, 2020) Karakus, Mahmut; Basar, Feyzi
Quite recently, the authors introduced the vector valued multiplier spaces associated to the series of bounded linear operators M-f(infinity)(Sigma(k) T-k) and M-wf(infinity)(Sigma(k) T-k) by means of almost and weak almost summability, respectively; [J. Math. Anal. Appl. 484: 123651]. As was recorded as an open problem in [J. Math. Anal. Appl. 484: 123651], in this study, we introduce vector valued multiplier spaces M-f lambda(infinity)(Sigma(k) T-k) and M-wf lambda(infinity)(Sigma(k) T-k) by means of generalized almost and weak almost summability, and give a characterization of completeness of these spaces, via c(0)(X)-multiplier convergent series. We also characterize the continuity and the (weak) compactness of the summing operator S from the multiplier spaces M-f lambda(infinity)(Sigma(k) T-k) or M-wf lambda(infinity)(Sigma(k) T-k) to an arbitrary normed space Y through c(0)(X)-multiplier Cauchy and l(infinity)(X)-multiplier convergent series, respectively. Finally, we prove that if Sigma(k) T-k is l(infinity)(X)-multiplier Cauchy, then the spaces M-f lambda(infinity)(Sigma(k) T-k) and M-wf lambda(infinity)(Sigma(k) T-k) are identical. These results are more general than the corresponding consequences given in [J. Math. Anal. Appl. 484: 123651] since almost convergence can be obtained from f(lambda) -convergence under certain conditions.