2018/01/29 by Nicola Thorn, Thorn, Nicola
Mathematics · #Holomorphic and Operator Theory #Approximation Theory and Sequence Spaces #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.1801.09478
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 \mathscrMf. We show that for 1 ≤ p ≤ q ≤ ∞, if f ∈ ℓr(ℚ+), then \mathscrMf: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 ‖\mathscrMf‖p,q = ‖f‖r,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 \mathscrMf 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