The problem addressed is the exact determination of the operator norm and lower bound of four-dimensional generalized Hausdorff matrices on the double sequence spaces $\mathcal{L}_p$. A Hardy type formulae is found as an exact value for their operator norm and a Copson type formulae is established as a lower estimate for their lower bound. Further, exact values are found for the operator norm and lower bound of the transpose of generalized Hausdorff matrices.
Gholamreza TALEBI
. OPERATOR NORM AND LOWER BOUND OF FOUR-DIMENSIONAL GENERALIZED HAUSDORFF MATRICES[J]. Acta mathematica scientia, Series B, 2022
, 42(1)
: 429
-436
.
DOI: 10.1007/s10473-022-0124-x
[1] Akgun F A, Rhoades B E. Upper bounds for E-J matrices. Appl Math Inf Sci, 2012, 3:1125-1128
[2] Başar F. Summability Theory and Its Applications. İstanbul:Bentham Science Publishers, 2012
[3] Başar F, Sever Y. The space $\mathcal{L}_p$ of double sequences. Math J Okayama Univ, 2009, 51:149-157
[4] Bennett G. Factorizing the classical inequalities. Mem Amer Math Soc, 1996, 120(576):1-130
[5] Chen C P, Wang K Z. Operator norms and lower bounds of generalized Hausdor matrices. Linear Multilinear Algebra, 2011, 59(3):321-337
[6] Endl K. Untersuchungen über Momentprobleme bei Verfahren vom Hausdorffschen Typus. Math Ann, 1960, 139:403-432
[7] Hardy G H. An inequality for Hausdorff means. J Lond Math Soc, 1943, 18:46-50
[8] Jakimovski A, Rhoades B E, Tzimbalario J. Hausdorff matrices as bounded operators over ℓp. Math Z, 1974, 138:173-181
[9] Rhoades B E. Some classes of doubly infinite matrices. Indian J Pure Appl Math, 2003, 23(6):419-427
[10] Talebi G. Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces. Linear Multilinear Algebra, 2017, 65(11):2257-2267
[11] Talebi G. Operator norm and lower bound of four-dimensional matrices. Indag Math, 2017, 28(6):1134- 1143
[12] Talebi G. Lower bound of four-dimensional Hausdorff matrices. J Inequal Appl, 2019, 2019(1):82, 1-7
[13] Talebi G. Complementary results on the boundedness problem of factorizable four-dimensional matrices. Bull Malays Math Sci Soc, 2020, 43(1):609-618
[14] Talebi G. Boundedness problem of four-dimensional matrices on the domain spaces of $\mathcal{L}_p$. Results Math, 2019, 74(1):1-16
[15] Yeşilkayagil M, Başar F. On the Domain of Riesz mean in the space $\mathcal{L}_s$. Filomat, 2017, 31(4):925-940
