Yeni $\Delta _{q}^{v}$-fark operatörü ve topolojik özellikleri

Öz

$\Delta_{q}^{v}$ fark operatörünü kullanarak $\Delta^{v}$’yi genişlettik. $l_{p} (\Delta_{q}^{v} )$ fark dizi uzayını oluşturduk ve bazı topolojik özelliklerini inceledik. Eğer $l_{p} (\Delta_{q}^{v} )$ uygun bir $\left\| . \right\|_{p,\Delta_{q}^{v} } $ normu verilirse bunun bir Banach uzayı olacağını gösterdik. Ayrıca $\left({l_{p} (\Delta_{q}^{v} ),\left\| . \right\|_{p,\Delta_{q}^{v} } } \right)$ ve $\left( {l_{p} ,\left\| . \right\|_{p} } \right)$ dizi uzaylarının lineer izometrik olduklarını gösterdik. Çalışmanın sonunda ise $l_{p}(\Delta _{q}^{v})\subset l_{p}\left( \cal M,\Delta _{q}^{v}\right) $ olduğu gösterildi. Orlicz fonksiyonlarının ailesi ${\cal M}$, $\Delta_{2}$ şartı ile örtüşmektedir.

Anahtar Kelimeler

• Fark dizi uzayları
• izometrik dizi uzayları
• dizi uzayları

New $\Delta _{q}^{v}$ -difference operator and topological features

Öz

We extented $\Delta^{v}$ by using difference operator $\Delta_{q}^{v}$. We generated the difference sequence space $l_{p} (\Delta_{q}^{v} )$ and investigated some of their properties. We showed that, if $l_{p} (\Delta_{q}^{v} )$ is supplied with an proper norm $\left\| . \right\|_{p,\Delta_{q}^{v} } $ then it will be a Banach space. We further more showed that, the sequence spaces $\left({l_{p} (\Delta_{q}^{v} ),\left\| . \right\|_{p,\Delta_{q}^{v} } } \right)$ and $\left( {l_{p} ,\left\| . \right\|_{p} } \right)$ are linearly isometric. At the end of this studies, it was shown that $l_{p}(\Delta _{q}^{v})\subset l_{p}\left( M,\Delta _{q}^{v}\right) $. The family of the Orlicz functions ${\cal M}$ is coincides the $\Delta_{2}$ condition.

Anahtar Kelimeler

• Difference sequence spaces
• isometric sequence spaces
• sequence spaces

Kaynakça

1. Kizmaz, H., On certain sequence spaces, Canadian Mathematical Bulletin, 24, 169-176, (1981).
2. Çolak, R. and Et, M., On some generalized difference sequence spaces and related matrix transformations, Hokkaido Mathematical Journal, 26, 3, 483-492, (1997).
3. Karakaş, A., Altın, Y. Et, M., On some topological properties of a new type difference sequence spaces, Advancements In Mathematical Sciences, Proceedings of the International Conference on Advancements in Mathematical Sciences (AMS-2015), Fatih University, Antalya, 144, (2015).
4. Karakaş, A., Altın,Y. and Çolak, R., On some topological properties of a new type difference sequence spaces, International Conference on Mathematics and Mathematics Education (ICMME-2016), Firat University, Elaziğ-Turkey, 167-168, (2016).
5. Peralta, Isometry of a sequence space generated by a difference operator, International Mathematical Forum, 5, 42, pp. 2077-2083, (2010).
6. Altay, B. and Polat, H., On some new Euler difference sequence spaces, Southeast Asian Bull. Mathematics., 30, 209-220, (2006).
7. Altay, B., Başar, F., The matrix domain and the fine spectrum of the difference operator on the sequence space , Communications in Mathematics and Applications, 2, 2, 1-11, (2007).
8. Qamaruddin and Saifi, A. H., Generalized difference sequence spaces defined by a sequence of Orlicz functions, Southeast Asian Bull. Mathematics, 29, 1125-1130, (2005).

9. Kamthan, P. K., Gupta, M., Sequence Spaces and Series, Marcel Dekker Inc. Newyork, (1981).
10. Krasnoselskii, M. A., and Rutickii, Y. B., Convex Functions and Orlicz Spaces, Groningen, Netherlands, (1961).
11. Kamthan, P. K., Convex functions and their applications, Journal of Istanbul University Faculty of Science. A Series, 28, 71-78, (1963).
12. Lindenstrauss, J. and Tzafriri, L., On Orlicz sequence spaces, Israel Journal. Mathematics, 10, 3, 379-390, (1971).