Bounded multiplicative Toeplitz operators on sequence spaces

الأصل · EN

In this paper, we study the linear mapping which sends the sequence x=(xₙ)ₙ ∈ ₙ to y=(yₙ)ₙ ∈ ₙ where yₙ = ∑ₖ₌₁∞ f(n/k)xₖ for f: Q+ → C. This operator is the multiplicative analogue of the classical Toeplitz operator, and as such we denote the mapping by Mf. We show that for 1 ≤ p ≤ q ≤ ∞, if f ∈ ℓʳ(Q+), then Mf:ℓᵖ → ℓq is bounded where 1/r = 1 - 1/p + 1/q. Moreover, for the cases when p=1 with any q, p=q, and q=∞ with any p, we find that the operator norm is given by Mfₚ,q = fᵣ,Q₊ when f ≥ 0. Finding a necessary condition and the operator norm for the remaining cases highlights an interesting connection between the operator norm of Mf and elements in ℓᵖ that have a multiplicative structure, when considering f:N → C. We also provide an argument suggesting that f ∈ ℓʳ may not be a necessary condition for boundedness when 1<p<q<∞.