BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESÀRO MATRIX∗

doi:10.1007/s10473-023-0404-0

摘要/Abstract

摘要： Let the triangle matrix $A^{ru}$ be a generalization of the Cesàro matrix and $U\in\{c_{0},c,\ell_{\infty}\}$. In this study, we essentially deal with the space $U(A^{ru})$ defined by the domain of $A^{ru}$ in the space $U$ and give the bases, and determine the Köthe-Toeplitz, generalized Köthe-Toeplitz and bounded-duals of the space $U(A^{ru})$. We characterize the classes $(\ell_{\infty}(A^{ru}):\ell_{\infty})$, $(\ell_{\infty}(A^{ru}):c)$, $(c(A^{ru}):c)$, and $(U:V(A^{ru}))$ of infinite matrices, where $V$ denotes any given sequence space. Additionally, we also present a Steinhaus type theorem. As an another result of this study, we investigate the $\ell_p$-norm of the matrix $A^{ru}$ and as a result obtaining a generalized version of Hardy's inequality, and some inclusion relations. Moreover, we compute the norm of well-known operators on the matrix domain $\ell_p(A^{ru})$.

关键词: matrix domain, normed sequence space, duals and matrix transformations

Abstract: Let the triangle matrix $A^{ru}$ be a generalization of the Cesàro matrix and $U\in\{c_{0},c,\ell_{\infty}\}$. In this study, we essentially deal with the space $U(A^{ru})$ defined by the domain of $A^{ru}$ in the space $U$ and give the bases, and determine the Köthe-Toeplitz, generalized Köthe-Toeplitz and bounded-duals of the space $U(A^{ru})$. We characterize the classes $(\ell_{\infty}(A^{ru}):\ell_{\infty})$, $(\ell_{\infty}(A^{ru}):c)$, $(c(A^{ru}):c)$, and $(U:V(A^{ru}))$ of infinite matrices, where $V$ denotes any given sequence space. Additionally, we also present a Steinhaus type theorem. As an another result of this study, we investigate the $\ell_p$-norm of the matrix $A^{ru}$ and as a result obtaining a generalized version of Hardy's inequality, and some inclusion relations. Moreover, we compute the norm of well-known operators on the matrix domain $\ell_p(A^{ru})$.

Key words: matrix domain, normed sequence space, duals and matrix transformations

Feyzi BA͒AR, Hadi ROOPAEI. BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESÀRO MATRIX^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1518-1536.

Feyzi BA͒AR, Hadi ROOPAEI. BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESÀRO MATRIX^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1518-1536.

参考文献

[1] Altay B, Ba͑ar F. Certain topological properties and duals of the domain of a triangle matrix in a sequence space. J Math Anal Appl,2007, 336(2): 632-645
[2] Altay B, Ba͑ar F, Mursaleen M. On the Euler sequence spaces which include the spaces $\ell_{p}$ and $\ell_{\infty}$ I.Inform Sci,2006, 176(10): 1450-1462
[3] Aydιn C, Ba͑ar F, On the new sequence spaces which include the spaces $c_{0}$ and $c$. Hokkaido Math J,2004, 33(2): 383-398
[4] Aydιn C, Ba͑ar F. Some new sequence spaces which include the spaces $\ell_{p}$ and $\ell_\infty$. Demonstratio Math,2005, 38(3): 641-656
[5] Ba͑ar F. Summability Theory and Its Applications. Boca Raton: CRC Press, 2022
[6] Ba͑ar F, Braha N L. Euler-Cesàro difference spaces of bounded, convergent and null sequences. Tamkang J Math,2006, 47(4): 405-420
[7] Ba͑ar F Dutta H. Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties. Boca Raton: CRC Press, 2020
[8] Ba͑ar F, Roopaei H. On the factorable spaces of absolutely $p$-summable, null, convergent and bounded sequences. Math Slovaca,2021, 71(6): 1375-1400
[9] Bennett G. Lower bounds for matrices II. Canad J Math, 1992, 44: 54-74
[10] Boos J.Classical and Modern Methods in Summability. New York: Oxford University Press, 2000
[11] Braha N L, Ba͑ar F. On the domain of the triangle $A(\lambda)$ on the spaces of null, convergent and bounded sequences. Abstr Appl Anal,2013, 2013: Art 476363
[12] Buntinas M. On Toeplitz sections in sequence spaces. Math Proc Cambridge Philos Soc, 1975, 78: 451-460
[13] Buntinas M, Tanoviɣ-Miller N. Absolute boundedness and absolute convergence in sequence spaces. Proc Amer Math Soc,1991, 111: 967-979
[14] Buntinas M, Tanoviɣ-Miller N. Strong boundedness and strong convergence in sequence spaces. Canad J Math,1991, 43: 960-974
[15] Chen C P, Luor D C, Ou Z Y. Extensions of Hardy inequality. J Math Anal Appl, 2002, 273: 160-171
[16] Cooke R G.Infinite Matrices and Sequence Spaces. London: Macmillan, 1950
[17] Fleming D J. Unconditional Toeplitz sections in sequence spaces. Math Z, 1987, 194: 405-414
[18] Foroutannia D, Roopaei H. The norms and the lower bounds for matrix operators on weighted difference sequence spaces. UPB Sci Bull Series A, 2017, 79(2): 151-160
[19] Gao P. A note on $\ell^p$ norms of weighted mean matrices. J Inequal Appl, 2012, : Art 110
[20] Garling D J H. On topological sequence spaces. Proc Cambridge Phil Soc, 1967, 63: 997-1019
[21] Grosse-Erdmann K G. On $\ell^1$-invariant sequence spaces. J Math Anal Appl, 2001, 262: 112-132
[22] Hardy G H. An inequality for Hausdorff means. J London Math Soc, 1943, 18: 46-50
[23] Hardy G H. Divergent Series.Oxford: Oxford University Press, 1973
[24] Hardy G H, Littlewood J E, Polya G.Inequalities. Cambridge: Cambridge University Press, 2001
[25] Ilkhan M. Norms and lower bounds of some matrix operators on Fibonacci weighted difference sequence space. Math Methods Appl Sci, 2019, 42(16): 5143-5153
[26] Jarrah A M, Malkowsky E. BK spaces, bases and linear operators. Rendiconti Circ Mat Palermo II, 1990, 52: 177-191
[27] Jarrah A M, Malkowsky E. Ordinary, absolute and strong summability and matrix transformations. Filomat, 2003, 17: 59-78
[28] Kamthan P K, Gupta M.Sequence Spaces and Series. New York: Marcel Dekker Inc, 1981
[29] Kiri͑çi M, Ba͑ar F. Some new sequence spaces derived by the domain of generalized difference matrix. Comput Math Appl,2010, 60(5): 1299-1309
[30] Maddox I J. On theorems of Steinhaus type. J London Math Soc, 1967, 42: 239-244
[31] Meyers G. On Toeplitz sections in $FK$-spaces. Studia Math, 1974, 51: 23-33
[32] Mursaleen M A. A note on matrix domains of Copson matrix of order $\alpha$ and compact operators. Asian-European J Math, 2022, 15(7): 2250140
[33] Mursaleen M. Applied Summability Methods. Boston: Springer, 2014
[34] Mursaleen M, Ba͑ar F. Sequence Spaces: Topics in Modern Summability Theor. Boca Raton: CRC Press, 2020
[35] Mursaleen M, Ba͑ar F, Altay B. On the Euler sequence spaces which include the spaces $\ell_{p}$ and $\ell_\infty$ I. Nonlinear Anal,2006, 65(3): 707-717
[36] Mursaleen M, Edely O. Compact operators on sequence spaces associated with the Copson matrix of order $\alpha$. J Inequal Appl, 2021, 2021: Art 178
[37] Mursaleen M, Roopaei H. Sequence spaces associated with fractional Copson matrix and compact operators. Results Math, 2021, 76: Art 134
[38] Roopaei H. Norms of summability and Hausdorff mean matrices on difference sequence spaces. Math Inequal Appl, 2019, 22(3): 983-987
[39] Roopaei H. Norm of Hilbert operator on sequence spaces. J Inequal Appl, 2020, 2020: Art 117
[40] Roopaei H. A study on Copson operator and its associated sequence space. J Inequal Appl, 2020, 2020: Art 120
[41] Roopaei H. A study on Copson operator and its associated sequence space II. J Inequal Appl, 2020, 2020: Art 239
[42] Roopaei H, Ba͑ar F. On the spaces of Cesàro absolutely $p$-summable null and convergent sequences. Math Methods Appl Sci,2021, 44(5): 3670-3685
[43] Roopaei H, Foroutannia D. The norms of certain matrix operators from $\ell_p$ spaces into $\ell_p(\Delta^n)$ spaces. Linear Multilinear Algebra, 2019, 67(4): 767-776
[44] Roopaei H, Foroutannia D, Ilkhan M, Kara E. Cesàro spaces and norm of operators on these matrix domains. Mediterr J Math,2020, 17: Art 121
[45] Sargent W. On sectionally bounded $BK$-spaces. Math Z, 1964, 83: 57-66
[46] Sember J J. On unconditional Toeplitz section boundedness in sequence spaces. Rocky Mountain J Math, 1977, 7: 699-706
[47] ͑engönül M, Ba͑ar F. Cesàro sequence spaces of non-absolute type which include the spaces $c_0$ and $c$. Soochow J Math,2005, 31(1): 107-119
[48] Sönmez A, Ba͑ar F. Generalized difference spaces of non-absolute type of convergent and null sequences. Abstr Appl Anal,2012, : Art 435076
[49] Stieglitz M, Tietz H. Matrix transformationen von folgenraumen eineergebnisübersicht. Math Z,1977, 154: 1-16
[50] Yaying T, Hazarika B, Mohiuddine S, Mursaleen M. Estimation of upper bounds of certain matrix operators on Binomial weighted sequence spaces. Adv Oper Theory, 2020, 5(4): 1376-1389
[51] Zeller K. Abschnittskonvergenz in FK-Ráumen. Math Z,1951, 55: 55-70

BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESÀRO MATRIX^∗

BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESÀRO MATRIX^∗

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