Multiplication operators between discrete Hardy spaces on rooted trees

Abstract

Muthukumar and Ponnusamy \cite{MP-Tp-spaces} studied the multiplication operators on ${\mathbb{T}}_p$ spaces. In this article, we mainly consider multiplication operators between ${\mathbb{T}}_p$ and ${\mathbb{T}}_q$ ($p\ne q$). In particular, we characterize bounded and compact multiplication operators from ${\mathbb{T}}_p$ to ${\mathbb{T}}_q$. For $p\ne q$, we prove that there are no invertible multiplication operators from ${\mathbb{T}}_p$ to ${\mathbb{T}}_q$ and also there are no isometric multiplication operators from ${\mathbb{T}}_p$ to ${\mathbb{T}}_q$. Finally, we discuss about fixed points of a multiplication operator on ${\mathbb{T}}_p$.

17 pages