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

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.

differentiation, discrete function spaces, infinite trees
Mathematical Subject Classification
Primary: 47B38
Secondary: 05C05
Received: 19 June 2021
Revised: 4 August 2021
Accepted: 5 August 2021
Published: 14 March 2022

Communicated by Stephan Garcia
Robert F. Allen
Department of Mathematics and Statistics
University of Wisconsin
La Crosse, WI
United States
Colin M. Jackson
Department of Mathematics
University of Colorado
Boulder, CO
United States