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A note on Lie product preserving maps on Mn(ℝ)

  • Janko Marovt EMAIL logo
Veröffentlicht/Copyright: 26. August 2016
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 66 Heft 3

Abstract

Let ϕ be an injective, continuous, Lie product preserving map on Mn(ℝ), n > 3. In the paper we show that then there exist an invertible matrix T ∈ Mn(ℝ) and a continuous function ψ:Mn(ℝ)→ ℝ, where ψ(A) = 0 for all matrices of trace zero, such that either ϕ(A) = TAT−1 + ψ(A)I for all A ∈ Mn(ℝ), or ϕ(A) = −TAtT−1 + ψ(A)I for all A ∈ Mn(ℝ). We determine that a similar proposition holds true for the set Mn(ℂ), n > 3.

MSC 2010: Primary 47B49; Secondary 15A04, 15A27, 17B60
Keywords: Lie product; non-linear preserver; commutativity

(Communicated by Werner Timmermann)

Acknowledgement

The author wishes to thank the referee for helpful comments and suggestions.

References

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Received: 2013-8-13
Accepted: 2013-10-28
Published Online: 2016-8-26
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0173
Schlagwörter für diesen Artikel
Lie product; non-linear preserver; commutativity

Artikel in diesem Heft

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
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  29. Research Article
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  31. Research Article
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  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
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  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
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  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
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