Document Type : Research Paper

Authors

• A. R. Moazzen
• R. Lashkaripour

Dept. of Math.,University of Sistan and Baluchestan , Zahedan, Iran.

Abstract

Let $A=(a_{n,k}{)}_{n,k\ge 1}$ and $B=(b_{n,k}{)}_{n,k\ge 1}$ be two non-negative matrices. Denote by $L_{v,p,q,B}(A)$, the supremum of those $L$, satisfying the following inequality:
$$\Vert Ax{\Vert }_{v,B(q)}\ge L\Vert x{\Vert }_{v,B(p)},$$
where $x\ge 0$ and $x\in l_p(v,B)$ and also$v=(v_n{)}_{n=1}^{\mathrm{\infty }}$ is an increasing, non-negative sequence of real numbers. In this paper, we obtain a Hardy-type formula for $L_{v,p,q,B}(H_{\mu })$, where $H_{\mu }$ is the Hausdorff matrix and $0<q\le p\le 1$. Also for the case $p=1$, we obtain $\Vert Ax{\Vert }_{v,B(1)}$, and for the case $p\ge 1$, we obtain $L_{v,p,q,B}(A)$.

Keywords

• Lower bound
• Weighted block sequence space
• Hausdorff matrices
• Euler matrices
• Cesaro matrices
• Matrix norm