Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence
In 2004–2006, the corresponding double sequence spaces were defined for the Pringsheim and the bounded Pringsheim convergence by Gokhan and Colak. In 2009, Colak and Mursaleen characterized some classes of matrix transformations transforming the space of bounded Pringsheim convergent (to 0) double s...
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2025-03-01
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| author | Maria Zeltser Şeyda Sezgek |
| author_facet | Maria Zeltser Şeyda Sezgek |
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| description | In 2004–2006, the corresponding double sequence spaces were defined for the Pringsheim and the bounded Pringsheim convergence by Gokhan and Colak. In 2009, Colak and Mursaleen characterized some classes of matrix transformations transforming the space of bounded Pringsheim convergent (to 0) double sequences with powers and the space of uniformly bounded double sequences with powers to the space of (bounded) Pringsheim convergent (to 0) double sequences. But many of their results appeared to be wrong. In 2024, we gave corresponding counterexamples and proved the correct results. Moreover, we gave the conditions for a wider class of matrices. As is well known, convergence of a double sequence in Pringsheim’s sense does not imply its boundedness. Assuming, in addition, boundedness for double sequences usually simplifies proofs. In this paper, we characterize matrix transformations transforming the space of Pringsheim convergent (to 0) double sequences with powers or the space of ultimately bounded double sequences with powers without assuming uniform boundedness. |
| format | Article |
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| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-3ae43ea200294f97a300d948ec4516da2025-08-20T02:42:22ZengMDPI AGMathematics2227-73902025-03-0113693010.3390/math13060930Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim ConvergenceMaria Zeltser0Şeyda Sezgek1Department of Mathematics, Tallinn University, Narva Mnt. 29, 10120 Tallinn, EstoniaDepartment of Mathematics, Mersin University, 33343 Mersin, TurkeyIn 2004–2006, the corresponding double sequence spaces were defined for the Pringsheim and the bounded Pringsheim convergence by Gokhan and Colak. In 2009, Colak and Mursaleen characterized some classes of matrix transformations transforming the space of bounded Pringsheim convergent (to 0) double sequences with powers and the space of uniformly bounded double sequences with powers to the space of (bounded) Pringsheim convergent (to 0) double sequences. But many of their results appeared to be wrong. In 2024, we gave corresponding counterexamples and proved the correct results. Moreover, we gave the conditions for a wider class of matrices. As is well known, convergence of a double sequence in Pringsheim’s sense does not imply its boundedness. Assuming, in addition, boundedness for double sequences usually simplifies proofs. In this paper, we characterize matrix transformations transforming the space of Pringsheim convergent (to 0) double sequences with powers or the space of ultimately bounded double sequences with powers without assuming uniform boundedness.https://www.mdpi.com/2227-7390/13/6/930matrix transformationMaddox sequence spacesdouble sequence |
| spellingShingle | Maria Zeltser Şeyda Sezgek Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence Mathematics matrix transformation Maddox sequence spaces double sequence |
| title | Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence |
| title_full | Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence |
| title_fullStr | Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence |
| title_full_unstemmed | Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence |
| title_short | Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence |
| title_sort | matrix transformations of double convergent sequences with powers for the pringsheim convergence |
| topic | matrix transformation Maddox sequence spaces double sequence |
| url | https://www.mdpi.com/2227-7390/13/6/930 |
| work_keys_str_mv | AT mariazeltser matrixtransformationsofdoubleconvergentsequenceswithpowersforthepringsheimconvergence AT seydasezgek matrixtransformationsofdoubleconvergentsequenceswithpowersforthepringsheimconvergence |