Bounded multiplicative Toeplitz operators on sequence spaces

In this paper, we study the linear mapping which sends the sequence $$x=\left(x_n\right)_{n\in\mathbb{N}}\;\mathrm{to}\;y=\left(y_n\right)_{n\in\mathbb{N}}\;\mathrm{where}\;y_n\;=\;\sum\nolimits^\infty_{k=1}f\left(n/k\right)x_k \;\mathrm{for}\;f:\mathbb{Q}^{+}\to\;\mathbb{C}.$$ This operator is the multiplicative analogue of the classical Toeplitz operator, and as such we denote the mapping by $$\mathcal{M}_f$$ . We show that for $$1\leq p\leq q\leq\infty,\;\mathrm{if}\;f\;\in l^r\left(\mathbb{Q}^{+}\right),\;\mathrm{then}\;\mathcal{M}_f\;:\;l^p\;\to\;l^q$$ is bounded where $$\frac{1}{r}\;=\;1-\frac{1}{p}\;+\;\frac{1}{q}.$$ Moreover, for the cases when p=1 with any $$q,\;p\;=\;q, \mathrm{and}\;q\;=\;\infty$$ with any p, we Find that the operator norm is given by $$\|\mathcal{M}_f\|_{p,q}\;=\;\|f\|_{r,\mathbb{Q}^{+}}\;\mathrm{When}\;f\geq 0.$$ Finding a necessary condition and the operator norm for the remaining cases highlights an interesting connection between the operator norm of $$\mathcal{M}_f$$ and elements in lp that have a multiplicative structure, when considering $$f\;:\;\mathbb{N}\;\to\;\mathbb{C}.$$ We also provide an argument suggesting that $$f\;\in\;l^{r}$$ may not be a necessary condition for boundedness when $$1<p<q<\infty$$ .