Some new Cesàro sequence spaces of order $\alpha$

Medine Yeşilkayagil Savaşcı; Feyzi Başar

doi:10.3842/umzh.v76i3.7333

Medine Yeşilkayagil Savaşcı Faculty of Applied Sciences, Uşak University, Turkey
Feyzi Başar Department of Primary Mathematics Teacher Education, İnönü University, Malatya, Turkey

DOI: https://doi.org/10.3842/umzh.v76i3.7333

Keywords: Normed sequence space, $\alpha$-, $\beta$- and $\gamma$-duals and matrix mappings

Abstract

UDC 517.98

We introduce the spaces $\ell_\infty(C_\alpha),$ $f(C_\alpha),$ and $f_0(C_\alpha)$ of Ces\`{a}ro bounded, Ces\`{a}ro almost convergent, and Ces\`{a}ro almost null sequences of order $\alpha,$ respectively. Moreover, we establish some inclusion relations for these spaces and determine the $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $\ell_\infty(C_\alpha)$ and $f(C_\alpha).$ Finally, we characterize the classes of matrix transformations from the space $f(C_\alpha)$ to any sequence space $Y$ and from any sequence space $Y$ to the space $f(C_\alpha).$

References

S. Banach, Theorie des operations lineaires, Warszawa (1932).

F. Baçsar, Strongly-conservative sequence-to-series matrix transformations, Erc. Üni. Fen Bil. Derg., 5, № 12, 888–893 (1989).

F. Baçsar, Summability theory and its applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton etc. (2022).

F. Başar, R. Çolak, Almost-conservative matrix transformations, Turk. J. Math., 13, № 3, 91–100 (1989).

F. Baçsar, I. Solak, Almost-coercive matrix transformations, Rend. Mat. Appl. (7), 11, № 2, 249–256 (1991).

F. Başar, M. Kirişçi, Almost convergence and generalized difference matrix, Comput. Math. Appl., 61, № 3, 602–611 (2011).

J. Boos, Classical and modern methods in summability, Oxford Univ. Press, New York (2000).

R. G. Cooke, Infinite matrices and sequence spaces, Machmillen and Co. Limited, London (1950).

R. çColak, Ö. Çakar, Banach limits and related matrix transformations, Stud. Sci. Math. Hung., 24, 429–436 (1989).

G. Das, Banach and other limits, J. London Math. Soc., 7, 501–507 (1973).

J. P. Duran, Infinite matrices and almost convergence, Math. Z., 128, 75–83 (1972).

A. M. Jarrah, E. Malkowsky, BK spaces, bases and linear operators, Rend. Circ. Mat. Palermo (2), 52, 177–191 (1990).

P. K. Kampthan, M. Gupta, Sequence spaces and series, Marcel Dekker Inc., New York, Basel (1981).

J. P. King, Almost summable sequences, Proc. Amer. Math. Soc., 17, 1219–1225 (1966).

G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80, 167–190 (1948).

I. J. Maddox, Elements of functional analysis, Cambridge Univ. Press, London (1970).

H. I. Miller, C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar., 93, 135–151 (2001).

E. Malkowsky, V. Rakocevic, Summability methods and applications, Mathematical Institute of the Serbian Academy of Sciences and Arts (2020).

M. Mursaleen, F. Baçsar, Sequence spaces: topics in modern summability theory, Ser. Math. and Appl., CRC Press, Taylor & Francis Group, Boca Raton etc. (2020).

G. M. Petersen, Regular matrix transformations, McGraw-Hill, New York etc. (1970).

A. Peyerimhoff, Lectures on summability, Lect. Notes in Math., vol. 107, Springer-Verlag, Berlin etc. (1969).

H. Roopaei, F. Baçsar, On the spaces of Cesàro absolutely $p$-summable, null, and convergent sequences, Math. Methods Appl. Sci., 44, № 5, 3670–3685 (2021).

J. A. Siddiqi, Infinite matrices summing every almost periodic sequences, Pacific J. Math., 39, № 1, 235–251 (1971).

M. Stieglitz, H. Tietz, Matrixtransformationen von Folgenräumen. Eine Ergebnisübersicht, Math. Z., 154, № 1, 1–16 (1977).

A. Wilansky, Summability through functional analysis, vol. 85, Mathematics Studies, North Holland, Amsterdam (1984).