  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 $\mathbb{T}_q$ ($p\neq q$).
In particular, we characterize bounded and compact multiplication operators
from $\mathbb{T}_{p}$ to $\mathbb{T}_{q}$. For $p\neq q$, 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 $\mathbb{T}_{p}$.
