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Comparison theorems for weak splittings of bounded operators

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Summary

Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Woźnicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.

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Authors and Affiliations

  1. Matematicko-Fyzikálni Fakulta, Katedra Numerické Matematiky, Univerzity Karlovy, Malostranské nám. 25, 11800, Praha 1, Czechoslovakia

    Ivo Marek

  2. Department of Mathematics, Temple University, 19122, Philadelphia, PA, USA

    Daniel B. Szyld

Authors
  1. Ivo Marek
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  2. Daniel B. Szyld
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This work was supported by the National Science Foundation grants DMS-8807338 and INT-8918502

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Marek, I., Szyld, D.B. Comparison theorems for weak splittings of bounded operators. Numer. Math. 58, 387–397 (1990). https://doi.org/10.1007/BF01385632

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  • DOI: https://doi.org/10.1007/BF01385632

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