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Roots in Operator and Banach Algebras

Integral Equations and Operator Theory volume 85, pages 63–90 (2016)Cite this article

Abstract

We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the ‘sign’ and ‘geometric mean’ of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    David P. Blecher & Zhenhua Wang

Authors
  1. David P. Blecher
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  2. Zhenhua Wang
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Corresponding author

Correspondence to David P. Blecher.

Additional information

In memoriam Charles Read—gentleman, brother, mathematical force of nature

Supported by a grant from the NSF.

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Cite this article

Blecher, D.P., Wang, Z. Roots in Operator and Banach Algebras. Integr. Equ. Oper. Theory 85, 63–90 (2016). https://doi.org/10.1007/s00020-015-2272-z

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  • DOI: https://doi.org/10.1007/s00020-015-2272-z

Mathematics Subject Classification

  • Primary 47A64
  • 47L10
  • 47L30
  • 47B44
  • Secondary 15A24
  • 15A60
  • 47A12
  • 47A60
  • 47A63
  • 49M15
  • 65F30

Keywords

  • Roots
  • fractional powers
  • geometric mean
  • sign of operator
  • Newton method for roots
  • binomial method for square root
  • accretive operator
  • sectorial operator
  • nonselfadjoint operator algebra
  • numerical range
  • functional calculus