## Banach Journal of Mathematical Analysis

### Geometric properties of the second-order Cesàro spaces

Naim L. Braha

#### Abstract

We prove that, for any $p\in(1,\infty)$, the second-order Cesàro sequence space $\operatorname {Ces}^{2}(p)$ has the $(\beta)$-property and the $k$-NUC property for $k\geq2$. In addition, we show that $\operatorname {Ces}^{2}(p)$ has the Kadec–Klee, rotundity, and uniform convexity properties. For any positive integer $k$, we also investigate the uniform Opial and $(L)$ properties of the sequence space. We also establish that $\operatorname {Ces}^{2}(p)$ is reflexive and has the fixed-point property. Finally, we calculate the packing constant $(C)$ of the space.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 1 (2016), 1-14.

Dates
Accepted: 20 March 2015
First available in Project Euclid: 15 October 2015

https://projecteuclid.org/euclid.bjma/1444913859

Digital Object Identifier
doi:10.1215/17358787-3158414

Mathematical Reviews number (MathSciNet)
MR3453520

Zentralblatt MATH identifier
1347.46012

#### Citation

Braha, Naim L. Geometric properties of the second-order Cesàro spaces. Banach J. Math. Anal. 10 (2016), no. 1, 1--14. doi:10.1215/17358787-3158414. https://projecteuclid.org/euclid.bjma/1444913859

#### References

• [1] N. L. Braha, Some geometric properties of $N(\lambda,p)$ spaces, J. Inequal. Appl. 2014, art. ID 112.
• [2] S. Chen, Geometry of Orlicz Spaces, Dissertationes Math. (Rozprawy Mat.) 356, Polish Acad. Sci. Inst. Math., Warsaw, 1996.
• [3] Y. Cui and H. Hudzik, On the uniform Opial property in some modular sequence spaces, Funct. Approx. Comment. Math. 26 (1998), 93–102.
• [4] Y. Cui, H. Hudzik, and R. Pluciennik, Banach–Saks property in some Banach sequence spaces, Ann. Polon. Math. 65 (1997), no. 2, 193–202.
• [5] Y. Cui, C. Meng, and R. Pluciennik, Banach–Saks property and property $(\beta)$ in Cesàro sequence spaces, Southeast Asian Bull. Math. 24 (2000), no. 2, 201–210.
• [6] J. Diestel, Geometry of Banach Spaces—Selected Topics, Springer, Berlin, 1984.
• [7] R. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990.
• [8] V. I. Gurariĭ, Differential properties of the convexity moduli of Banach spaces (in Russian), Mat. Issled. 2 (1967), 141–148.
• [9] C. A. Kottman, Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150 (1970), 565–576.
• [10] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford 18 (1967), 345–355.
• [11] H. Nakano, Modulared sequence spaces, Proc. Japan Acad. 27 (1951), 508–512.
• [12] H. Nergiz and F. Basar, Some geometric properties of the domain of the double sequential band matrix $B(\tilde{r},\tilde{s})$ in the sequence space $l(p)^{*}$, Abstr. Appl. Anal. 2013, art. ID 949282.
• [13] S. Prus, Banach spaces with uniform Opial property, Nonlinear Anal. 8 (1992), no. 8, 697–704.
• [14] E. Savaş, V. Karakaya, and N. Şimşek, Some $l(p)$-type new sequence spaces and their geometric properties, Abstr. Appl. Anal. 2009, art. ID 696971.
• [15] S. Simons, The sequence spaces $l(p_{v})$ and $m(p_{v})$, Proc. London Math. Soc. 15 (1965), 422–436.
• [16] J. S. Shiue, On the Cesàro sequence space, Tamkang J. Math. 1 (1970), no. 1, 19–25.