Vol. 15, No. 1, 2022

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The differentiation operator on discrete function spaces of a tree

Robert F. Allen and Colin M. Jackson

Vol. 15 (2022), No. 1, 163–184
Abstract

We study the differentiation operator acting on discrete function spaces, that is, spaces of functions defined on an infinite rooted tree. We discuss, through its connection with composition operators, the boundedness and compactness of this operator. In addition, we discuss the operator norm and spectrum and consider when such an operator can be an isometry. We then apply these results to the operator acting on the discrete Lipschitz space and weighted Banach spaces, as well as the Hardy spaces defined on homogeneous trees.

Keywords
differentiation, discrete function spaces, infinite trees
Primary: 47B38
Secondary: 05C05