Lower bound and upper bound of operators on block weighted sequence spaces

Rahmatollah Lashkaripour; Gholomraza Talebi

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 293-304
  • ISSN: 0011-4642

Abstract

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Let A = ( a n , k ) n , k 1 be a non-negative matrix. Denote by L v , p , q , F ( A ) the supremum of those L that satisfy the inequality A x v , q , F L x v , p , F , where x 0 and x l p ( v , F ) and also v = ( v n ) n = 1 is an increasing, non-negative sequence of real numbers. If p = q , we use L v , p , F ( A ) instead of L v , p , p , F ( A ) . In this paper we obtain a Hardy type formula for L v , p , q , F ( H μ ) , where H μ is a Hausdorff matrix and 0 < q p 1 . Another purpose of this paper is to establish a lower bound for A W N M v , p , F , where A W N M is the Nörlund matrix associated with the sequence W = { w n } n = 1 and 1 < p < . Our results generalize some works of Bennett, Jameson and present authors.

How to cite

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Lashkaripour, Rahmatollah, and Talebi, Gholomraza. "Lower bound and upper bound of operators on block weighted sequence spaces." Czechoslovak Mathematical Journal 62.2 (2012): 293-304. <http://eudml.org/doc/246225>.

@article{Lashkaripour2012,
abstract = {Let $A=(a_\{n,k\})_\{n,k\ge 1\}$ be a non-negative matrix. Denote by $L_\{v,p,q,F\}(A)$ the supremum of those $L$ that satisfy the inequality \[ \Vert Ax\Vert \_\{v,q,F\} \ge L\Vert x\Vert \_\{v,p,F\}, \] where $x\ge 0$ and $x\in l_p(v,F)$ and also $v=(v_n)_\{n=1\}^\infty $ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_\{v,p,F\}(A)$ instead of $L_\{v,p,p,F\}(A)$. In this paper we obtain a Hardy type formula for $L_\{v,p,q,F\}(H_\mu )$, where $H_\mu $ is a Hausdorff matrix and $0<q\le p\le 1$. Another purpose of this paper is to establish a lower bound for $\Vert A_\{W\}^\{NM\} \Vert _\{v,p,F\}$, where $A_\{W\}^\{NM\}$ is the Nörlund matrix associated with the sequence $W=\lbrace w_n\rbrace _\{n=1\}^\infty $ and $1<p<\infty $. Our results generalize some works of Bennett, Jameson and present authors.},
author = {Lashkaripour, Rahmatollah, Talebi, Gholomraza},
journal = {Czechoslovak Mathematical Journal},
keywords = {lower bound; weighted sequence space; Hausdorff matrices; Euler matrices; Cesàro matrices; Hölder matrices; Gamma matrices; weighted sequence space; Hausdorff matrix; Euler matrix; Cesàro matrix; Hölder matrix; Gamma matrix},
language = {eng},
number = {2},
pages = {293-304},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lower bound and upper bound of operators on block weighted sequence spaces},
url = {http://eudml.org/doc/246225},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Lashkaripour, Rahmatollah
AU - Talebi, Gholomraza
TI - Lower bound and upper bound of operators on block weighted sequence spaces
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 293
EP - 304
AB - Let $A=(a_{n,k})_{n,k\ge 1}$ be a non-negative matrix. Denote by $L_{v,p,q,F}(A)$ the supremum of those $L$ that satisfy the inequality \[ \Vert Ax\Vert _{v,q,F} \ge L\Vert x\Vert _{v,p,F}, \] where $x\ge 0$ and $x\in l_p(v,F)$ and also $v=(v_n)_{n=1}^\infty $ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_{v,p,F}(A)$ instead of $L_{v,p,p,F}(A)$. In this paper we obtain a Hardy type formula for $L_{v,p,q,F}(H_\mu )$, where $H_\mu $ is a Hausdorff matrix and $0<q\le p\le 1$. Another purpose of this paper is to establish a lower bound for $\Vert A_{W}^{NM} \Vert _{v,p,F}$, where $A_{W}^{NM}$ is the Nörlund matrix associated with the sequence $W=\lbrace w_n\rbrace _{n=1}^\infty $ and $1<p<\infty $. Our results generalize some works of Bennett, Jameson and present authors.
LA - eng
KW - lower bound; weighted sequence space; Hausdorff matrices; Euler matrices; Cesàro matrices; Hölder matrices; Gamma matrices; weighted sequence space; Hausdorff matrix; Euler matrix; Cesàro matrix; Hölder matrix; Gamma matrix
UR - http://eudml.org/doc/246225
ER -

References

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  7. Foroutannia, D., Upper bound and lower bound for matrix operators on weighted sequence space, PhD. Thesis Zahedan (2007). (2007) 
  8. Jameson, G. J. O., Lashkaripour, R., Norms of certain operators on weighted l p spaces and Lorentz sequence spaces, JIPAM, J. Inequal. Pure Appl. Math. 3 (2002), Electronic only. (2002) Zbl1021.47019MR1888921
  9. jun., P. D. Johnson, Mohapatra, R. N., Ross, D., 10.1090/S0002-9939-96-03081-X, Proc. Am. Math. Soc. 124 (1996), 543-547. (1996) Zbl0846.40007MR1301506DOI10.1090/S0002-9939-96-03081-X
  10. Lashkaripour, R., Foroutannia, D., 10.1007/s10587-009-0006-6, Czech. Math. J. 59 (134) (2009), 81-94. (2009) Zbl1217.47065MR2486617DOI10.1007/s10587-009-0006-6
  11. Lashkaripour, R., Foroutannia, D., 10.1134/S1995080209010065, Lobachevskii J. Math. 30 (2009), 40-45. (2009) Zbl1177.26039MR2506053DOI10.1134/S1995080209010065
  12. Lashkaripour, R., Talebi, G., Lower bound of Copson type for Hausdorff matrices on weighted sequence spaces, J. Sci., Islam. Repub. Iran 22 (2011), 153-157. (2011) MR2884149
  13. Lashkaripour, R., Talebi, G., 10.7153/jmi-05-04, J. Math. Inequal. 5 (2011), 33-38. (2011) Zbl1211.26018MR2799056DOI10.7153/jmi-05-04
  14. Lashkaripour, R., Talebi, G., Bounds for the operator norms of some Nörlund matrices on weighted sequence spaces, Preprint. 

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