To define a new sequence space and determine the Köthe-Toeplitz duals of this sequence space, characterizing the matrix transformation classes between the defined sequence spaces and classical sequence spaces has been an important area of work for researchers. Defining and examining a new vector-valued sequence space is also a considerable field of study since it generalizes classical sequence spaces. In this study, new vector-valued sequence spaces $ E(X, \lambda, p) $ are introduced. The Köthe-Toeplitz duals of $ E(X, \lambda, p) $ spaces are identified. Also, necessary and sufficient conditions are determined for $ A = (A_{nk}) $ to belong to the matrix classes $ (E(X, \lambda, p), c(q)) $; where $ A_{ nk}\in B(X, Y) $, $ X\in \{c, \ell_\infty\} $ and $ Y $ is any Banach spaces.
Citation: Osman Duyar. On some new vector valued sequence spaces $ E(X, \lambda, p) $[J]. AIMS Mathematics, 2023, 8(6): 13306-13316. doi: 10.3934/math.2023673
To define a new sequence space and determine the Köthe-Toeplitz duals of this sequence space, characterizing the matrix transformation classes between the defined sequence spaces and classical sequence spaces has been an important area of work for researchers. Defining and examining a new vector-valued sequence space is also a considerable field of study since it generalizes classical sequence spaces. In this study, new vector-valued sequence spaces $ E(X, \lambda, p) $ are introduced. The Köthe-Toeplitz duals of $ E(X, \lambda, p) $ spaces are identified. Also, necessary and sufficient conditions are determined for $ A = (A_{nk}) $ to belong to the matrix classes $ (E(X, \lambda, p), c(q)) $; where $ A_{ nk}\in B(X, Y) $, $ X\in \{c, \ell_\infty\} $ and $ Y $ is any Banach spaces.
| [1] | N. Rath, Operator duals of some sequence spaces, Indian J. Pure Appl. Math., 20 (1989), 953–963. |
| [2] | J. K. Srivastava, B. K. Srivastava, Generalized sequence space $c_0(X, \lambda, p)$, Indian J. Pure Appl. Math., 27 (1996), 73–84 |
| [3] | B. Altay, F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30 (2006), 591–608. |
| [4] | M. Mursaleen, A. K. Noman, On the spaces of $\lambda$-convergent and bounded sequences, Thai J. Math., 8 (2012), 311–329. |
| [5] | I. J. Maddox, Spaces of strongly sequences, Q. J. Math., 18 (1967), 345–355, |
| [6] | I. J. Maddox, Infinite matrices of operators, Springer, 1980. |
| [7] |
S. Suantai, C. Sudsukh, Matrix transformations of the Maddox vector-valued sequence spaces, Southeast Asian Bull. Math., 26 (2003), 337–350. http://doi.org/10.1007/s100120200055 doi: 10.1007/s100120200055
|
| [8] |
S. Suantai, Matrix transformations of the some vector-valued sequence spaces, Southeast Asian Bull. Math., 24 (2000), 297–303. http://doi.org/10.1007/s10012-000-0297-x doi: 10.1007/s10012-000-0297-x
|
| [9] |
S. Suantai, W. Sanhan, On $\beta$-dual of vector-valued sequence spaces of Maddox, Int. J. Math. Math. Sci., 30 (2001), 385–392. http://doi.org/10.1155/S0161171202012772 doi: 10.1155/S0161171202012772
|
| [10] |
K. Goswin, G. Erdmann, The structure of the sequence spaces of Maddox, Can. J. Math., 44 (1992), 298–307. http://doi.org/10.4153/CJM-1992-020-2 doi: 10.4153/CJM-1992-020-2
|
| [11] | C. X. Wu, L. Liu, Matrix transformation on some vector-valued sequence spaces, SEA Bull. Math., 17 (1993), 83–96. |
Osman Duyar. On some new vector valued sequence spaces $ E(X, \lambda, p) $[J]. AIMS Mathematics, 2023, 8(6): 13306-13316. doi: 10.3934/math.2023673
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