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Integral Equations and Operator Theory

, Volume 85, Issue 1, pp 63–90 | Cite as

Roots in Operator and Banach Algebras

  • David P. BlecherEmail author
  • Zhenhua Wang
Article
  • 165 Downloads

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.

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 

Mathematics Subject Classification

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

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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

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