1 Bounds for matrices on weighted sequence spaces

By D. Foroutannia.

abstract Let w = (w_{n}) be a decreasing non-negative sequence and F be a partition of positive integers. If F = (F_{n}), where each F_{n} is a finite interval of positive integers and also for all n, max F_{n} < min F_{n+1}. The block weighted sequence space l_{p}(w, F) is the space of all real sequences x = (x_{n}) with

∥x∥_{p,w,F} = (∑_{n=1}^{∞} w_{n}| < x, F_{n} > |^{p})^{1/p} < ∞,

where < x, F_{n} >= ∑_{i∈F_{n}} x_{i}. In this paper, we consider inequalities of the form ∥Ax∥_{p,w,F} ≤ L∥Bx∥_{q,v,F}, where A and B are matrix operators, x decreasing non-negative sequence and w, v are weights and also F is a block. Moreover, this study is an extension of some works of which are studied before on sequence spaces l_{p}(v) by J. Pecaric, I. Peric and R. Roki in [3].