Previous |  Up |  Next

Article

Keywords:
inequality; norm; summability matrix; Hausdorff matrix; Nörlund matrix; weighted mean matrix; weighted sequence space and Lorentz sequence space
Summary:
In this paper we consider the problem of finding upper bounds of certain matrix operators such as Hausdorff, Nörlund matrix, weighted mean and summability on sequence spaces $l_p(w)$ and Lorentz sequence spaces $d(w,p)$, which was recently considered in [9] and [10] and similarly to [14] by Josip Pecaric, Ivan Peric and Rajko Roki. Also, this study is an extension of some works by G. Bennett on $l_p$ spaces, see [1] and [2].
References:
[1] G. Bennett: Factorizing the classical inequalities. Mem. Amer. Math. Soc. 576 (1996), 1–130. MR 1317938 | Zbl 0857.26009
[2] G. Bennett: Inequalities complimentary to Hardy. Quart. J. Math. Oxford (2) 49 (1998), 395–432. MR 1652236 | Zbl 0929.26013
[3] D. Borwein and F. P. Cass: Nörlund matrices as bounded operators on $l_p$. Arch. Math. 42 (1984), 464–469. DOI 10.1007/BF01190697 | MR 0756700
[4] D. Borwein: Nörlund operators on $l_p$. Canada. Math. Bull. 36 (1993), 8–14. DOI 10.4153/CMB-1993-002-x | MR 1205888
[5] G. H. Hardy: An inequality for Hausdorff means. J. London Math. Soc. 18 (1943), 46–50. MR 0008854 | Zbl 0061.12704
[6] G. H. Hardy: Divergent Series. 2nd edition, American Mathematical Society, 2000.
[7] G. H. Hardy and J. E. Littlewood: A maximal theorem with function-theoretic. Acta Math. 54 (1930), 81–116. DOI 10.1007/BF02547518 | MR 1555303
[8] G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities. 2nd edition, Cambridge University press, Cambridge, 2001.
[9] G. J. O. Jameson and R. Lashkaripour: Lower bounds of operators on weighted $l_p$ spaces and Lorentz sequence spaces. Glasgow Math. J. 42 (2000), 211–223. DOI 10.1017/S0017089500020061 | MR 1763740
[10] G. J. O. Jameson and R. Lashkaripour: Norms of certain operators on weighted $l_p$ spaces and Lorentz sequence spaces. J. Inequalities in Pure and Applied Mathematics, 3, Issue 1, Article 6 (2002). MR 1888921
[11] R. Lashkaripour: Lower bounds and norms of operators on Lorentz sequence spaces. Doctoral dissertation (Lancaster, 1997).
[12] R. Lashkaripour: Transpose of the Weighted Mean operators on Weighted Sequence Spaces. WSEAS Transaction on Mathematics, Issue 4, 4 (2005), 380–385.
[13] R. Lashkaripour and D. Foroutannia: Lower Bounds for Matrices on Weighted Sequence Spaces. Journal of Sciences Islamic Republic of IRAN, 18 (2007), 49–56. MR 2499829
[14] J. Pecaric, I. Peric and R. Roki: On bounds for weighted norms for matrices and integral operators. Linear Algebra and Appl. 326 (2001), 121–135. MR 1815954
Partner of
EuDML logo