Bounded multiplicative Toeplitz operators on sequence spaces

TL;DR

This paper investigates bounded multiplicative Toeplitz operators on sequence spaces, establishing conditions for boundedness and operator norms, and exploring the connection between these operators and multiplicative structures in sequences.

Contribution

It introduces a multiplicative analogue of Toeplitz operators, providing boundedness criteria, explicit operator norms, and insights into the role of $ ext{ell}^r$ conditions for these operators.

Findings

Boundedness of $ ext{M}_f$ from $ ext{ell}^p$ to $ ext{ell}^q$ when $f ext{ in } ext{ell}^r$

Operator norm equals $ ext{ell}^r$ norm of $f$ for specific cases

Potential necessity of $ ext{ell}^r$ condition for boundedness remains uncertain

Abstract

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 $1<p<q<\infty$.