Bounded multiplicative Toeplitz operators on sequence spaces

In this paper, we study the linear mapping which sends the sequence $x=(x_n)_{n \in \mathbb{N}}$ to $y=(y_n)_{n \in \mathbb{N}}$ where $y_n = \sum_{k=1}^\infty f(n/k)x_k$ 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 $\mathscr{M}_f$. We show that for $1 \leq p \leq q \leq \infty$, if $f \in \ell^r(\mathbb{Q}^+)$, then $\mathscr{M}_f:\ell^p \to \ell^q$ is bounded where $\frac{1}{r} = 1 - \frac{1}{p} + \frac{1}{q}$. Moreover, for the cases when $p=1$ with any $q$, $p=q$, and $q=\infty$ with any $p$, we find that the operator norm is given by $\|\mathscr{M}_f\|_{p,q} = \|f\|_{r,\mathbb{Q}^+}$ 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 $\mathscr{M}_f$ and elements in $\ell^p$ that have a multiplicative structure, when considering $f:\mathbb{N} \to \mathbb{C}$. We also provide an argument suggesting that $f \in \ell^r$ may not be a necessary condition for boundedness when $1<p<q<\infty$.