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

Authors

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

Abstract

Let A = (an;k)n;k1 and B = (bn;k)n;k1 be two non-negative ma-
trices. Denote by Lv;p;q;B(A), the supremum of those L, satisfying the following
inequality:
k Ax kv;B(q) L k x kv;B(p);
where x  0 and x 2 lp(v;B) and also v = (vn)1n
=1 is an increasing, non-negative
sequence of real numbers. In this paper, we obtain a Hardy-type formula for
Lv;p;q;B(H), where H is the Hausdor matrix and 0 < q  p  1. Also for the
case p = 1, we obtain kAkw;B(1), and for the case p  1, we obtain Lw;B(p)(A).

Keywords