Yazar "Karakus, Mahmut" seçeneğine göre listele

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    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.
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    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.
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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.
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    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.
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    Vector valued multiplier spaces of f?-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.