Abstract
In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesàro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesàro summability.
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Aizpuru, A., Pérez-Fernández, F.J.: Characterizations of series in Banach spaces. Math. Univ. Comen. 58(2), 337–344 (1999)
Aizpuru, A., Gutiérrez-Dávila, A., Sala, A.: Unconditionally Cauchy series and Cesàro summability. J. Math. Anal. Appl. 324, 39–48 (2006)
Altay, B., Başar, F.: Certain topological properties and duals of the domain of a triangle matrix in a sequence space. J. Math. Anal. Appl. 336(2), 632–645 (2007)
Başar, F.: Summability Theory and Its Applications. Bentham Science Publishers, İstanbul (2012)
Bessaga, C., Pelczynski, A.: On bases and unconditional convergence of series in Banach spaces. Stud. Math. 17, 151–164 (1958)
Kama, R., Altay, B.: Weakly unconditionally Cauchy series and Fibonacci sequence spaces. J. Inequal. Appl. 2017, 133 (2017). doi:10.1186/s13660-017-1407-y
Kama, R., Altay, B.: Some sequence spaces and completeness of normed spaces. Creat. Math. Inform., (to appear)
Ng, P.-N., Lee, P.-Y.: Cesàro sequences spaces of non-absolute type. Comment. Math. Prace Mat. 20(2), 429–433 (1978)
Pérez-Fernández, F.J., Benítez-Trujillo, F., Aizpuru, A.: Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series. Czechoslov. Math. J. 50(125), 889–896 (2000)
Ronglu, L., Qingying, B.: Locally convex spaces containing no copy of \(c_0\). J. Math. Anal. Appl. 172, 205–211 (1993)
Swartz, C.: Operator valued series and vector valued multiplier spaces. Casp. J. Math. Sci. 3(2), 277–288 (2014)
Swartz, C.: Multiplier Convergent Series. World Scientific Publishing, Singapore (2009)
Swartz, C.: Orlicz–Pettis theorems for multiplier convergent operator valued series. Proyecciones 22(2), 135–144 (2003)
Şengönül, M., Başar, F.: Some new Cesàro sequence spaces of non-absolute type which include the spaces \(c_0\) and \(c\). Soochow J. Math. 31(1), 107–119 (2005)
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The authors wish to thank the referee for his/her valuable suggestions, which improved the paper considerably.
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Altay, B., Kama, R. On Cesàro summability of vector valued multiplier spaces and operator valued series. Positivity 22, 575–586 (2018). https://doi.org/10.1007/s11117-017-0528-3
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DOI: https://doi.org/10.1007/s11117-017-0528-3
Keywords
- Multiplier convergent series
- Cesàro sequence spaces
- Summing operator
- Compact operators
- Orlicz–Pettis theorem