On a Class of Vector-Valued Sequences Associated with Multiplier Sequences

Abstract

In this article we introduce the vector valued sequence space m(Ek ,φ ,Λ), associated with the multiplier sequence Λ = (λk) of non-zero complex numbers, and the terms of the sequence are chosen from the seminormed spaces Ek ,s eminormed byfk for all k ∈ N. This generalizes the sequence space m(φ) introduced and studied by Sargent(10). We study some of its properties like solidity, completeness, and obtain some inclusion results. We also characterize the multiplier problem and obtain the corresponding spaces dual to m(Ek ,φ ,Λ). We prove some general results too. Throughout the article Ek, k ∈ N , are seminormed spaces, seminormed by fk's and X is a seminormed space, seminormed by q .A lsow(Ek) ,c (Ek) ,� ∞(Ek) ,� p(Ek) denote the spaces of all, convergent, bounded and p-absolutely summable Ek-valued sequences respectively. For Ek = C, the field of complex numbers, for all k ∈ N , these represent the corresponding scalar- valued sequence spaces. The zero elements of Ek's will be denoted by θk; the zero sequence is denoted by θ =( θk). The sequence space m(φ) was introduced by Sargent (10) , who studied some of its properties and obtained its relationship with the spacep. Later on it was investigated from the sequence space point of view and linked with the summability by Rath and Tripathy (8) , Tripathy (13) , Tripathy and Sen (14) and others. The scope for the studies on sequence spaces was extended by using the notion of associ- ated multiplier sequences. Goes et al. (3) defined the differentiated sequence space dE and the integrated sequence space E for a given sequence space E, using multiplier sequences (k −1 ) and (k) respectively. Kamthan (5) used the multiplier sequence (k!). In the present article we shall consider a general multiplier sequence Λ = (λk) of non-zero scalars. Let Λ = (λk) be a sequence of non-zero scalars. Then for a sequence space E, the multiplier sequence space E(Λ), associated with the multiplier sequence Λ is defined as