J ournal of Mathematical I nequalities Volume 5, Number 1 (2011), 33–38 LOWER BOUND FOR THE NORM OF LOWER TRIANGULAR MATRICES ON BLOCK WEIGHTED SEQUENCE SPACES R. LASHKARIPOUR AND G. TALEBI (Communicated by R. Oinarov) Abstract. Let 1 < p < ∞ and A =(a n,k ) n,k1 be a non-negative matrix. Denote by ‖A‖ w, p,F , the infimum of those U satisfying the following inequality: ‖Ax‖ w, p,F  U ‖x‖ w, p,I , where x  0 and x ∈ l p (w, I ) and also w =(w n ) ∞ n=1 is a decreasing, non-negative sequence of real numbers. The purpose of this paper is to give a lower bound for ‖A‖ w, p,F , where A is a lower triangular matrix. In particular, we apply our results to Weighted mean matrices and N¨ orlund matrices which recently considered in [2,3,6] on the usual sequence spaces. Our results generalize some work of Jameson, Lashkaripour, Frotannia and Chen in [4,7,8]. 1. Introduction Let p  1 and (w n ) ∞ n=1 be a decreasing, non-negative sequence of real numbers. We define the weighted sequence space l p (w) as l p (w)=  x =(x k ) : ∞ ∑ k=1 w k |x k | p < ∞  , with a norm ‖.‖ w, p which is defined in the following way: ‖x‖ w, p :=  ∞ ∑ k=1 w k |x k | p  1 p . Next, assume that F is a partition of positive integers. If F =(F n ) , where each (F n ) is a finite interval of positive integers and max F n < min F n+1 (n = 1, 2, 3, ...), we define the weighted sequence space l p (w , F ) as l p (w , F )=  x =(x k ) : ∞ ∑ k=1 w k |〈 x, F k 〉| p < ∞  , Mathematics subject classification (2010): 26D15, 47A30, 40G05, 47D37, 46A45, 54D55. Keywords and phrases: Norm, upper bound, lower triangular matrix, N ¨ orlund matrices, weighted mean matrices, block weighted sequence space. c  , Zagreb Paper JMI-05-04 33