Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces

Feyzi Başar

doi:10.1501/Commua1_0000000685

EN

Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces

Abstract

In the present paper, we summarize the literature on the normed and paranormed sequence spaces derived by the domain of the matrix lambda. Moreover, we establish some inclusion relations concerning with those spaces and determine their alpha-, beta- and gamma-duals. Finally, we record some open problems and further suggestions related with A summability.

Keywords

Normed and paranormed sequence spaces, matrix domain, trianglematrices, lambda matrix, almost convergence, alpha-, beta- and gamma-duals, matrix transformations

References

B. Altay, On the space of p-summable diğ erence sequences of order m, (1 p < 1), Stud. Sci. Math. Hungar. 43(4)(2006), 387–402.
B. Altay, F. Ba¸sar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26(5)(2002), 701–715.
B. Altay, F. Ba¸sar, Some Euler sequence spaces of non-absolute type, Ukrainian Math. J. 57(1)(2005), 1–17.
B. Altay, F. Ba¸sar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 30(5)(2006), 591–608.
B. Altay, F. Ba¸sar, Some paranormed sequence spaces of non-absolute type derived by weighted mean, J. Math. Anal. Appl. 319(2)(2006), 494–508.
B. Altay, F. Ba¸sar, Generalization of the sequence space `(p) derived by weighted mean, ibid. 330(1)(2007), 174–185.
B. Altay, F. Ba¸sar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, ibid. 336(1)(2007), 632–645.
B. Altay, F. Ba¸sar, The matrix domain and the …ne spectrum of the diğ erence operator on the sequence space `p, (0 < p < 1), Commun. Math. Anal. 2(2)(2007), 1–11.

C. Aydın, F. Ba¸sar, On the new sequence spaces which include the spaces c0and c, Hokkaido Math. J. 33(2)(2004), 383–398.
C. Aydın, F. Ba¸sar, Some new paranormed sequence spaces, Inform. Sci. 160(1-4)(2004), 27–40.
C. Aydın,F. Ba¸sar,Some new diğ erence sequence spaces, Appl. Math. Comput. 157(3)(2004), 677–693.
C. Aydın, F. Ba¸sar, Some new sequence spaces which include the spaces `pand `1, Demon- stratio Math. 38(3)(2005), 641–656.
C. Aydın, F. Ba¸sar, Some generalizations of the sequence space ar, Iran. J. Sci. Technol. p Trans. A, Sci. 30(2006), No. A2, 175–190.
F. Ba¸sar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, ·Istanbul-2012.
F. Ba¸sar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55(1)(2003), 136–147.
F. Ba¸sar, B. Altay, M. Mursaleen, Some generalizations of the space bvpof p-bounded vari- ation sequences, Nonlinear Anal. 68(2)(2008), 273–287.
M. Ba¸sarır, On some new sequence spaces and related matrix transformations, Indian J. Pure Appl. Math. 26(10)(1995), 1003–1010.
M. Ba¸sarır, On the generalized Riesz B-diğ erence sequence spaces, Filomat 24(4)(2010), 35–52.
M. Ba¸sarır, M. Kayıkçı, On the generalized Bm-Riesz sequence space and -property, J. Inequal. Appl. 2009 (2009), Article ID 385029, 18 pp.
M. Ba¸sarır, M. Öztürk, On the Riesz diğ erence sequence space, Rend. Circ. Mat. Palermo (2)57(2008), no. 3, 377–389.
E.E. Kara, M. Ba¸sarır, On compact operators and some Euler B(m)-diğ erence sequence spaces, J. Math. Anal. Appl. 379 (2011) 499–511.
E.E. Kara, M. Öztürk, M. Ba¸sarır, Some topological and geometric properties of generalized Euler sequence spaces, Math. Slovaca 60(3)(2010), 385–398.
V. Karakaya, A.K. Noman, H. Polat, On paranormed Math. Comput. Model. 54(2011), 1473–1480.
H. Kızmaz, On certain sequence spaces, Canad. Math. Bull. 24(2)(1981), 169–176.
G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190.
I.J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2), 18(1967), 345–355.
I.J. Maddox, Paranormed sequence spaces generated by in…nite matrices, Proc. Camb. Phil. Soc. 64(1968), 335–340.
I.J. Maddox, Some properties of paranormed sequence spaces, London J. Math. Soc. (2)1(1969), 316–322.
E. Malkowsky, Recent results in the theory of matrix transformations in sequence spaces, Mat. Vesnik 49(1997), 187–196.
E. Malkowsky, Mursaleen, S. Suantai, The dual spaces of sets of diğ erence sequences of order m and matrix transformations, Acta Math. Sin. Eng. Ser. 23(3)(2007), 521–532.
E. Malkowsky, V. Rakoµcevi´c, Measure of noncompactness of linear operators between spaces of sequences that are (N ; q) summable or bounded, Czechoslovak Math. J. 51(126) (2001), no. 3, 505–522.
E. Malkowsky, V. Rakoµcevi´c, S. ¼Zivkovi´c, Matrix transformations between the sequence spaces 0( ), v 0wp( ), vp( ), c0( ) (1 < p < 1) and certain BK spaces, Appl. Math. Comput. 147 (2) (2004), 377–396.
E. Malkowsky, E. Sava¸s, Matrix transformations between sequence spaces of generalized weighted means, Appl. Math. Comput. 147(2)(2004), 333–345. [34] F. Móricz, On
strong convergence of numerical sequences and Fourier series, Acta Math. Hung. 54(3-4)(1989), 319–327.
M. Mursaleen, A. Alotaibi, Statistical summability and approximation by de la Vallée-Poussin mean, Appl. Math. Lett. 24(3)(2011), 320–324.
M. Mursaleen, A.K. Noman, On the spaces of convergent and bounded sequences, Thai J. Math. 8(2)(2010), 311–329.
M. Mursaleen, A.K. Noman, On some new diğ erence sequence spaces of non-absolute type, Math. Comput. Modelling 52(3-4)(2010), 603–617.
M. Mursaleen, A.K. Noman, Applications of the Hausdorğ measure of noncompactness in some sequence spaces of weighted means, Comput. Math. Appl. 60(5)(2010), 1245–1258.
M. Mursaleen, A.K. Noman, The Hausdorğ measure of noncompactness of matrix operators on some BK spaces, Oper. Matrices 5(3)(2011), 473–486.
M. Mursaleen, A.K. Noman, On some new sequence spaces of non-absolute type related to the spaces `pand `1I, Filomat 25(2011), 33–51.
M. Mursaleen, A.K. Noman, On some new sequence spaces of non-absolute type related to the spaces `pand `1II, Math. Commun. 16(2011), 383–398.
M. Mursaleen, A.K. Noman, Compactness of matrix operators on some new diğ erence se- quence spaces, Linear Algebra Appl. 436(1)(2012), 41–52.
P.-N. Ng, P.-Y. Lee, Cesàro sequence spaces of non-absolute type, Comment. Math. Prace Mat. 20(2)(1978), 429–433.
H. Polat, F. Ba¸sar, Some Euler spaces of diğ erence sequences of order m, Acta Math. Sci. Ser. B Engl. Ed. 27B(2)(2007)
S. Simons, The sequence spaces `(pv) and m(pv), Proc. London Math. Soc. (3), 15(1965), 422–436.
A. Sönmez, F. Ba¸sar, Generalized diğ erence spaces of non-absolute type of convergent and null sequences, Abstr. Appl. Anal. 2012, Art. ID 435076, 20 pp.
M. ¸Sengönül, F. Ba¸sar, Some new Cesàro sequence spaces of non-absolute type which include the spaces cand c, Soochow J. Math. 31(1)(2005), 107–119.
C.-S. Wang, On Nörlund sequence spaces, Tamkang J. Math. 9(1978), 269–274.
M. Ye¸silkayagil, F. Ba¸sar, On the …ne spectrum of the operator de…ned by a lambda matrix over the sequence space c0and c, AIP Conference Proceedings 1470(2012), 199–202.
M. Ye¸silkayagil, F. Ba¸sar, Spaces of almost lambda null and almost lambda convergent se- quences, under communication.
S. Zeren, Ç.A. Bekta¸s, On some new sequence spaces of non-absolute type, submitted to: Allahabad Math. Soc.
Current address : Feyzi Ba¸sar;Fatih University, Faculty of Arts and Sciences, Department of Mathematics, The Hadımköy Campus, Büyükçekmece, 34500–·Istanbul, Turkey
E-mail address : fbasar@fatih.edu.tr, feyzibasar@gmail.com
URL: http://communications.science.ankara.edu.tr/index.php?series=A1

Details

Primary Language

English

Subjects

-

Journal Section

-

Authors

Feyzi Başar This is me

Publication Date

February 1, 2013

Submission Date

-

Acceptance Date

-

Published in Issue

Year 2013 Volume: 62 Number: 1

DOI

https://doi.org/10.1501/Commua1_0000000685

IZ

https://izlik.org/JA85HH54XM

Cite

RIS / Bibtex

APA

Başar, F. (2013). Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 62(1), 45-59. https://doi.org/10.1501/Commua1_0000000685

AMA

1.Başar F. Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2013;62(1):45-59. doi:10.1501/Commua1_0000000685

Chicago

Başar, Feyzi. 2013. “Survey on the Domain of the Matrix Lambda in the Normed and Paranormed Sequence Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 62 (1): 45-59. https://doi.org/10.1501/Commua1_0000000685.

EndNote

Başar F (February 1, 2013) Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 62 1 45–59.

IEEE

[1]F. Başar, “Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 62, no. 1, pp. 45–59, Feb. 2013, doi: 10.1501/Commua1_0000000685.

ISNAD

Başar, Feyzi. “Survey on the Domain of the Matrix Lambda in the Normed and Paranormed Sequence Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 62/1 (February 1, 2013): 45-59. https://doi.org/10.1501/Commua1_0000000685.

JAMA

1.Başar F. Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2013;62:45–59.

MLA

Başar, Feyzi. “Survey on the Domain of the Matrix Lambda in the Normed and Paranormed Sequence Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 62, no. 1, Feb. 2013, pp. 45-59, doi:10.1501/Commua1_0000000685.

Vancouver

1.Feyzi Başar. Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2013 Feb. 1;62(1):45-59. doi:10.1501/Commua1_0000000685

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