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A note on the surjectivity of operators on vector bundles over discrete spaces

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Abstract

In this note we give a short and self-contained proof for a criterion of Eidelheit on the solvability of linear equations in infinitely many variables. We use this criterion to study the surjectivity of magnetic Schrödinger operators on bundles over graphs.

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Notes

  1. This comes from the observation that initial states which are not in the image of a given cellular automaton can never be attained after iterating it. A garden of Eden theorem is then a theorem that provides criteria for the nonexistence of such states, that is, criteria for the surjectivity of the automaton.

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Acknowledgements

The authors are grateful to thank Daniel Lenz for pointing out the problem to them. Moreover, M.S. thanks Jürgen Voigt for an interesting discussion on closed range theorems.

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  1. Mathematisches Institut, Friedrich Schiller Universität Jena, 07743, Jena, Germany

    Jannis Koberstein & Marcel Schmidt

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  1. Jannis Koberstein
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  2. Marcel Schmidt
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Correspondence to Marcel Schmidt.

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Koberstein, J., Schmidt, M. A note on the surjectivity of operators on vector bundles over discrete spaces. Arch. Math. 114, 313–329 (2020). https://doi.org/10.1007/s00013-019-01412-8

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