The Riesz Core of a Sequence

Abstract

 The Riesz sequence space q

cr including the space c has recently been defined in [14] and its some properties
have been investigated. In the present paper, we introduce a new type core, Kq-core, of a complex valued
sequence and also determine the required conditions for a matrix B for which Kq-core (Bx) ⊆ K-core (x), Kqcore (Bx) ⊆ stA-core (x) and Kq-core (Bx) ⊆ Kq-core (x) hold for all x ∈ ∞ A . 

Keywords

• Matrix transformations
• core of a sequence
• statistical convergence

References

1. Abdullah M. Alotaibi, “Cesáro statistical core of complex number sequences”, Inter. J. Math. Math. Sci., Article ID 29869 (2007).
2. B. Altay, F. Başar, “Some paranormed Riesz sequence spaces of non-absolute type”, Southeast Asian Bull. Math. 30(5): 591-608 (2006).
3. F. Başar, “A note on the triangle limitation methods”, Fırat Univ. Fen & Müh. Bil. Dergisi, 5(1): 113-117 (1993).
4. R. G. Cooke, “Infinite matrices and sequence spaces”, Macmillan, New York (1950).
5. J. Connor, “On strong matrix summability with respect to a modulus and statistical convergence”, Canad. Math. Bull. 32: 194-198 (1989).
6. C. Çakan, H. Çoşkun, “Some new inequalities related to the invariant means and uniformly bounded function sequences”, Applied Math. Lett. 20(6): 605-609 (2007).
7. H. Çoşkun, C. Çakan, “A class of statistical and σ- conservative matrices”, Czechoslovak Math. J. 55(3): 791-801 (2005).
8. H. Çoşkun, C. Çakan, Mursaleen, “On the statistical and σ –cores”, Studia Math. 154(1):(2003).

9. K. Demirci, “A-statistical core of a sequence”, Demonstratio Math., 33: 43-51 (2000).
10. H. Fast, “Sur la convergence statisque”, Colloq. Math., 2: 241-244 (1951).
11. A. R. Freedman, J. J. Sember, “Densities and summability”, Pasific J. Math., 95:293-305 (1981).
12. J. A. Fridy, C. Orhan, “Statistical core theorems”, J. Math. Anal. Appl., 208: 520-527 (1997).
13. I. J. Maddox, “Elements of Functional Analysis”, Cambridge University Press, Cambridge (1970).
14. E. Malkowsky, V. Rakoćević, “Measure of noncompactness of linear operators between spaces of sequences that are (N, q) summable or bounded”, Czechoslovac Math. J., 51(126): 505- 522 (2001).
15. G. M. Petersen, “Regular matrix transformations”, McGraw-Hill, (1966).
16. A. A. Shcherbakov, “Kernels of sequences of complex numbers and their regular transformations”, Math. Notes, 22: 948-953 (1977).