ON MULTIPLIER WEIGHTED-SPACE OF SEQUENCES

Abstract

We consider the weighted spaces `p(S, φ) and `p(S, ψ) for 1 < p < +∞, where φ and ψ are weights on S (= N or Z). We obtain a sufficient condition for `p(S, ψ) to be multiplier weighted-space of `p(S, φ) and `p(S, ψ). Our condition characterizes the last multiplier weightedspace in the case where S = Z. As a consequence, in the particular case where ψ = φ, the weighted space `p(Z,ψ) is a convolutive algebra. 1. Preliminaries and introduction Let S (S = N or S = Z) and p ∈ ]1,+∞[. We say that ω is a weight on S if ω : S −→ [1,+∞[, is a map satisfying: ∑ n∈S ω(n) 1 1−p < +∞. We consider the weighted space: `(S, ω) = { (a(n))n∈S ∈ C S : ∑ n∈S |a(n)| ω(n) < +∞ } . Endowed with the norm |·|p,ω defined by: |a|p,ω = (∑ n∈S |a(n)| ω(n) ) 1 p for every (a(n))n∈S ∈ ` (S, ω), the space `(S,ω) becomes a Banach subspace of ` (S). We say that the weight ω is m-convolutive if a positive constant γ = γ(ω) exists such that: ω 1 1−p ∗ ω 1 1−p ≤ γ ω 1 1−p , where ∗ denotes the convolution product. If a= (a(n))n∈S ∈ `(S, ω), we define the complex function F(a) by F(a)(t) = ∑ n∈S a(n)e for every t ∈ R. Received February 6, 2020; Revised April 8, 2020; Accepted July 2, 2020. 2010 Mathematics Subject Classification. 46J10, 46H30.