  We introduce a class of iterated logarithmic Lipschitz spaces
$\mathcal{L}^{(k)}$, $k\in\mathbb{N}$, on an infinite tree which arise
naturally in the context of operator theory. We characterize boundedness and
compactness of the multiplication operators on $\mathcal{L}^{(k)}$ and provide
estimates on their operator norm and their essential norm. In addition, we
determine the spectrum, characterize the multiplication operators that are
bounded below, and prove that on such spaces there are no nontrivial isometric
multiplication operators and no isometric zero divisors.

Functional Analysis