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How Small Can a Sum of Idempotents Be?

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Abstract

The issue discussed in this paper is: how small can a sum of idempotents be? Here smallness is understood in terms of nilpotency or quasinilpotency. Thus the question is: given idempotents \(p_1,\ldots ,p_n\) in a complex algebra or Banach algebra, is it possible that their sum \(p_1+\cdots +p_n\) is quasinilpotent or (even) nilpotent (of a certain order)? The motivation for considering this problem comes from earlier work by the authors on the generalization of the logarithmic residue theorem from complex function theory to higher (possibly infinite) dimensions.

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Acknowledgements

The second author (T.E.) was supported by the Simons Foundation Collaboration Grant # 525111.

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Authors and Affiliations

  1. Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR, Rotterdam, The Netherlands

    Harm Bart

  2. Mathematics Department, University of California, Santa Cruz, CA, 95064, USA

    Torsten Ehrhardt

  3. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Bernd Silbermann

Authors
  1. Harm Bart
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  2. Torsten Ehrhardt
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  3. Bernd Silbermann
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Correspondence to Torsten Ehrhardt.

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Bart, H., Ehrhardt, T. & Silbermann, B. How Small Can a Sum of Idempotents Be?. Integr. Equ. Oper. Theory 92, 10 (2020). https://doi.org/10.1007/s00020-020-2566-7

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  • DOI: https://doi.org/10.1007/s00020-020-2566-7

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