In 1967, I. J. Maddox generalized the spaces c0, c, ℓ∞ by adding the powers pk (k ∈ ℕ) in the definitions of the spaces to the terms of elements of sequences (xk). Gökhan and Çolak in 2004–2006 defined the corresponding double sequence spaces for the Pringsheim and the bounded Pringsheim convergence. We will additionally define the corresponding double sequence spaces for the regular convergence. In 2009, Gökhan, Çolak and Mursaleen characterized some classes of matrix transformations involving these double sequence spaces with powers. However, many of their results appeared to be wrong. In this paper, we give corresponding counterexamples and prove the correct results. Moreover, we present the conditions for a wider class of matrices.
1. Başar, F. and Sever, Y. The space ℒq of double sequences. Math. J. Okayama Univ., 2009, 51, 149–157.
2. Boos, J. Classical and Modern Methods in Summability. Oxford University Press, Oxford, 2000.
https://doi.org/10.1093/oso/9780198501657.001.0001
3. Boos, J. and Seydel, D. Theorems of Toeplitz–Silverman type for maps defined by sequences of matrices. J. Anal., 2001, 9, 149–181.
4. Gökhan, A. and Çolak, R. The double sequence spaces c2P (p) and c2PB (p). Appl. Math. Comput., 2004, 157(2), 491–501.
https://doi.org/10.1016/j.amc.2003.08.047
5. Gökhan, A. and Çolak, R. Double sequence space ℓ2∞(p). Appl. Math. Comput., 2005, 160, 147–153.
https://doi.org/10.1016/j.amc.2003.08.142
6. Gökhan, A. and Çolak, R. On double sequence spaces 0c2P (p), 0c2PB (p) and ℓ2(p). Int. J. Pure Appl. Math., 2006, 30(3), 309–321.
7. Gökhan, A., Çolak, R. and Mursaleen, M. Some matrix transformations and generalized core of double sequences. Math. Comput. Model., 2009, 49(7–8), 1721–1731.
https://doi.org/10.1016/j.mcm.2008.12.002
8. Grosse-Erdmann, K.-G. Matrix transformations between the sequence spaces of Maddox. J. Math. Anal. Appl., 1993, 180(1), 223–238.
https://doi.org/10.1006/jmaa.1993.1398
9. Lascarides, C. G. and Maddox, I. J. Matrix transformations between some classes of sequences. Math. Proc. Cambridge Philos. Soc., 1970, 68, 99–104.
https://doi.org/10.1017/S0305004100001109
10. Maddox, I. J. Spaces of strongly summable sequences, Quarterly J. Math., 1967, 18(1), 345–355.
https://doi.org/10.1093/qmath/18.1.345
11. Nakano, H. Modulared sequence spaces. Proc. Japan Acad., 1951, 27(9), 508–512.
https://doi.org/10.3792/pja/1195571225
12. Zeltser, M., Boos, J. and Leiger, T. Sequences of 0’s and 1’s: new results via double sequence spaces. J. Math. Anal. Appl., 2002, 275(2), 883–899.
https://doi.org/10.1016/S0022-247X(02)00444-4
13. Zeltser, M. Investigation of double sequence spaces by soft and hard analytical methods. PhD thesis. University of Tartu, Estonia, 2001.
