Toeplitz operator and the operator ∇ on the [formula omitted] space of a tree

Abstract

Let T be a rooted, countable infinite tree without terminal vertices. Suppose that the degrees of the vertices of T are bounded. In this paper, we obtain a new upper bound for the norm of the operator ∇ on the Lp space of T. Moreover, we establish several sufficient conditions for Toeplitz operators to be compact on Lp with 1⩽p⩽∞. This answers two questions concerning the norm estimate of ∇ and the compactness of Toeplitz operators posed by Colonna and Martínez-Avendaño (2017) [10].