Some inequalities involving lower bounds of operators on weighted sequence spaces by a matrix norm

Let A=(a_{n,k})_{n,k\geq۱} and B=(b_{n,k})_{n,k\geq۱} be two non-negative matrices. Denote by L_{v,p,q,B}(A), the supremum of those L, satisfying the following inequality:\|Ax\|_{v,B(q)}\geq L\|x\|_{v,B(p)},where x\geq ۰ and x \in l_p(v,B) and alsov = (v_n)_{n=۱}^\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 ۰ < q \leq p \leq۱. Also for the case p = ۱, we obtain \|Ax\|_{v,B(۱)}, and for the case p\geq ۱, we obtain L_{v,p,q,B}(A).

کلیدواژه ها:

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