On Zweier generalized difference ideal convergent sequences in a locally convex space defined by Musielak-Orlicz function

Bipan Hazarika,Karan Tamanag

doi:10.5269/bspm.v35i2.29077

Abstract

Let $\mathbf{M}=(M_k)$ be a Musielak-Orlicz function. In this article, we introduce a new class of ideal convergent sequence spaces defined by Musielak-Orlicz function, using an infinite matrix, and a generalized difference matrix operator $B_{(i)}^{p}$ in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We obtainsome relations related to these sequence spaces.

Highlights

Throughout the article w, l∞, c, c0, denote for the classes of all, bounded, convergent, null sequences of complex numbers, respectively
∆0(i)xk = xk for all k ∈ N, which is equivalent to the following binomial representation: p
B(0i)xk = xk for all k ∈ N, which is equivalent to the following binomial representation: p

Summary

The space lM with the norm

≤1 k=1 becomes a Banach space which is called an Orlicz sequence space. A sequence M = (Mk) of Orlicz functions is called a Musielak-Orlicz function (for details see [10,13,17,20]). A Musielak-Orlicz function φ = (φk) is called a complementary function of a Musielak-Orlicz function M if φk(t) = sup{| t|s − Mk(s) : s ≥ 0}, for k = 1, 2, 3,. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space lM and its subspace hM are defined as follows: lM = {x = (xk) ∈ w : IM(cx) < ∞, for some c > 0}; hM = {x = (xk) ∈ w : IM(cx) < ∞, for all c > 0}, where IM is a convex modular defined by IM = Mk(xk), x = (xk) ∈ lM

We consider lM equipped with the Luxemburg norm

Thus we have

Also we have

Mk t ρ uk