Harun POLAT

Some New Cauchy Sequence Spaces

In this paper, our goal is to introduce some new Cauchy sequence spaces. These spaces are defined by Cauchy transforms. We shall use notations $C_{\infty }\left( s,t\right) $, $C\left( s,t\right) $ and $C_{0}\left( s,t\right) ~$for these new sequence spaces. We prove that these new sequence spaces $C_{\infty }\left( s,t\right) $, $C\left( s,t\right) $ and $C_{0}\left( s,t\right) ~$ are the $BK-$spaces and isomorphic to the spaces $l_{\infty }$, $c\ $and $c_{0}$, respectively. Besides the bases of these spaces, $\alpha -$, $\beta -\ $and $\gamma -$ duals of these spaces will be given. Finally, the matrix classes $(C\left( s,t\right) :l_{p})$ and $(C\left( s,t\right) :c)$ have been characterized.

Keywords:

Cauchy sequence spaces, $\alpha -$, $~\beta -\ $and $% \gamma -$ duals, Schauder basis Matrix mappings,

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[1] B. Choudhary, S. Nanda, Functional analysis with applications, Wiley, New Delhi, 1989.
[2] M. Kirişçi, On the Taylor sequence spaces of non-absolute type which include the spaces c0 and c, J. Math. Anal., 6(2) (2015) 22-35.
[3] B. Altay, F. Başar, Some Euler sequence spaces of non-absolute type, Ukrain. Math. J., 57(1) (2005), 1–17.
[4] B. Altay, F. Başar, M. Mursaleen, On the Euler sequence spaces which include in the spaces lp and l¥, Inform. Sci., 176(10) (2006), 1450–1462.
[5] E. Malkowsky, Recent results in the theory of matrix transformations in sequences spaces, Mat. Vesnik, 49 (1997), 187–196.
[6] P. N. Ng, P. Y. Lee, Cesaro sequences spaces of non-absolute type, Comment. Math. Prace Mat., 20(2) (1978), 429–433.
[7] C. S. Wang, On Nörlund seqence spaces, Tamkang J. Math., 9 (1978), 269-274.
[8] M. Candan, Domain of the double sequential band matrix in the spaces of convergent and null sequences, Adv. Dif. Equ., 1 (2014), 163-281.
[9] M. Candan, Almost convergence and double sequential band matrix, Acta. Math. Sci., 34B(2) (2014), 354-366.
[10] B. Altay, On the space of p-summable difference sequences of order $(1\leq p<\infty )$, Studia Sci. Math. Hungar., 43(4) (2006), 387–402.
[11] B. Altay, F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336(1) (2007), 632–645.
[12] C. Aydın,F. Başar, On the new sequence spaces which include the spaces $c_{0}$ and $c$, Hokkaido Math. J., 33(2) (2004), 383–398.
[13] F. Başar, Strongly-conservative sequence-to-series matrix transformations, Erc. Uni. Fen Bil. Derg., 5(12) (1989), 888–893.
[14] F. Başar, f–conservative matrix sequences, Tamkang J. Math., 22(2) (1991), 205–212.
[15] F. Başar, E. Malkowsky, B. Altay, Matrix transformations on the matrix domains of triangles in the spaces of strongly C1-summable and bounded sequences, Publ. Math., 73(1-2) (2008), 193–213.
[16] H. Polat, F. Başar, Some Euler spaces of difference sequences of order m, Acta Math. Sci., 27B(2) (2007), 254–266.
[17] M. S¸engçnül, F. Başar, Some new Cesaro sequence spaces of non-absolute type which include the spaces c0 and c, Soochow J. Math., 31(1) (2005), 107–119.
[18] P. V. Krishna Raja and at all, A cryptosystem based on Hilbert matrix using Cipher block chaining mode, Int. J. Math. Trends Tech., July to Aug Issue 2011.
[19] M. Mohammad Tabanjeh, New approach for the inversion of structured matrices via Newton’s iteration, Adv. Linear Algebra Matrix Theory, 5 (2015), 1-15.
[20] I. J. Maddox, Elements of functional analysis, Cambridge University Press, Cambridge, 1988.
[21] D. J. H. Garling, The $\alpha -$,$~\beta -\ $and $\gamma -$ duality of sequence spaces, Proc. Comb. Phil. Soc., 63 (1967), 963-981.
[22] M. Stieglitz, H. Tietz, Matrixtransformationen von folgenraumen eine ergebnisübersict, Math. Z., 154 (1977), 1-16.
[23] M. Candan, A new sequence space isomorphic to the space $l(p)$ and compact operators, J. Math. Comput. Sci., 4(2) (2014), 306-334.