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

Lashkaripour, Rahmatollah ; Talebi, Gholomraza

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 293-304.

Résumé

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 $$ \|Ax\|_{v,q,F} \ge L\| x\|_{v,p,F}, $$ where $x\ge 0$ and $x\in l_p(v,F)$ and also $v=(v_n{)}_{n=1}^{\mathrm{\infty }}$ 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 $0$. 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 $W=\{w_n{\}}_{n=1}^{\mathrm{\infty }}$ and $1$. Our results generalize some works of Bennett, Jameson and present authors.

Détail

DOI : 10.1007/s10587-012-0031-8

Classification : 26D15, 40G05, 46A45, 47A30, 54D55
Mots-clés : lower bound; weighted sequence space; Hausdorff matrices; Euler matrices; Cesàro matrices; Hölder matrices; Gamma matrices