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The Forward and Backward Shift on the Lipschitz Space of a Tree

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Abstract

We initiate the study of the forward and backward shifts on the Lipschitz space of an undirected tree, \(\mathcal {L}\), and on the little Lipschitz space of an undirected tree, \(\mathcal {L}_0\). We determine that the forward shift is bounded both on \(\mathcal {L}\) and on \(\mathcal {L}_0\) and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift, we determine when it is bounded on \(\mathcal {L}\) and on \(\mathcal {L}_0\), we find the norm when the tree is homogeneous, we calculate the spectrum for the case when the tree is homogeneous, and we determine, for a general tree, when it is hypercyclic.

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Notes

  1. We thank a referee for suggesting this question and for providing the reference in the previous paragraph.

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Authors and Affiliations

  1. Departamento Académico de Matemáticas, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

    Rubén A. Martínez-Avendaño

  2. Centro de Investigación en Matemáticas, Guanajuato, Mexico

    Emmanuel Rivera-Guasco

Authors
  1. Rubén A. Martínez-Avendaño
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  2. Emmanuel Rivera-Guasco
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Correspondence to Rubén A. Martínez-Avendaño.

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We would like to thank the reviewers for their suggestions, which greatly improved this paper. The first author’s research is partially supported by the Asociación Mexicana de Cultura A.C.

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Martínez-Avendaño, R.A., Rivera-Guasco, E. The Forward and Backward Shift on the Lipschitz Space of a Tree. Integr. Equ. Oper. Theory 92, 3 (2020). https://doi.org/10.1007/s00020-019-2558-7

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