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Transformations Preserving Norms of Means of Positive Operators and Nonnegative Functions

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Abstract

Motivated by recent investigations on norm-additive and spectrally multiplicative maps on various spaces of functions, in this paper we determine all bijective transformations between the positive cones of standard operator algebras over a Hilbert space which preserve a given symmetric norm of a given mean of elements. A result of similar spirit is also presented concerning transformations between cones of nonnegative elements of certain algebras of continuous functions.

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References

  1. Abatzoglou T.J.: Norm derivatives on spaces of operators. Math. Ann. 239, 129–135 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Busch P., Gudder S.P.: Effects as functions on projective Hilbert spaces. Lett. Math. Phys. 47, 329–337 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chan J.T., Li C.K., Tu C.C.N.: A class of unitarily invariant norms on B(H). Proc. Am. Math. Soc. 129, 1065–1076 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hatori O., Jiménez-Vargas A., Villegas-Vallecillos M.: Maps which preserve norms of non-symmetrical quotients between groups of exponentials of Lipschitz functions. J. Math. Anal. Appl. 415, 825–845 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hatori O., Hirasawa G., Miura T.: Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras. Cent. Eur. J. Math. 8, 597–601 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hatori, O., Lambert, S., Luttman, A., Miura, T., Tonev, T., Yates, R.: Spectral preservers in commutative Banach algebras. In: Jarosz, K. (ed.) Function Spaces in Modern Analysis. Contemporary Mathematics, vol. 547, pp. 103–124. American Mathematical Society, Providence (2011)

  7. Hosseini M., Font J.J.: Norm-additive in modulus maps between function algebras. Banach J. Math. Anal. 8, 79–92 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hosseini M., Sady F.: Maps between Banach function algebras satisfying certain norm conditions. Cent. Eur. J. Math. 11, 1020–1033 (2013)

    MATH  MathSciNet  Google Scholar 

  9. Jiménez-Vargas A., Lee K., Luttman A., Villegas-Vallecillos M.: Generalized weak peripheral multiplicativity in algebras of Lipschitz functions. Cent. Eur. J. Math. 11, 1197–1211 (2013)

    MATH  MathSciNet  Google Scholar 

  10. Kubo F., Ando T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lee K.: Characterizations of peripherally multiplicative mappings between real function algebras. Publ. Math. Debr. 84, 379–397 (2014)

    Article  MATH  Google Scholar 

  12. Molnár L.: Order-automorphisms of the set of bounded observables. J. Math. Phys. 42, 5904–5909 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Molnár L.: Some characterizations of the automorphisms of B(H) and C(X). Proc. Am. Math. Soc. 130, 111–120 (2002)

    Article  MATH  Google Scholar 

  14. Molnár L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics, vol. 1895. Springer, Berlin (2007)

    Google Scholar 

  15. Molnár L.: Maps preserving the geometric mean of positive operators. Proc. Am. Math. Soc. 137, 1763–1770 (2009)

    Article  MATH  Google Scholar 

  16. Molnár L.: Maps preserving the harmonic mean or the parallel sum of positive operators. Linear Algebra Appl. 430, 3058–3065 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Molnár L.: Maps preserving general means of positive operators. Electron. J. Linear Algebra 22, 864–874 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Molnár, L.: General Mazur–Ulam type theorems and some applications. Operator Theory: Advances and Applications (to appear)

  19. Nagy, G.: Preservers for the p-norm of linear combinations of positive operators. Abstr. Appl. Anal. 2014, 434121-1–434121-9 (2014)

  20. Rao, N.V., Tonev, T.V., Toneva, E.T.: Uniform algebra isomorphisms and peripheral spectra. In: Malios A., Haralampidou, M. (eds.) Topological Algebras. Contemporary Mathematics, vol. 427, pp. 401–416 (2007)

  21. Šemrl P.: Comparability preserving maps on bounded observables. Integral Equ. Oper. Theory 62, 441–454 (2008)

    Article  MATH  Google Scholar 

  22. Šemrl P.: Symmetries on bounded observables: a unified approach based on adjacency preserving maps. Integral Equ. Oper. Theory 72, 7–66 (2012)

    Article  MATH  Google Scholar 

  23. Tonev T.: Spectral conditions for almost composition operators between algebras of functions. Proc. Am. Math. Soc. 142, 2721–2732 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tonev T., Yates R.: Norm-linear and norm-additive operators between uniform algebras. J. Math. Anal. Appl. 357, 45–53 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. MTA-DE “Lendület” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, P.O. Box 12, Debrecen, 4010, Hungary

    Lajos Molnár & Patrícia Szokol

Authors
  1. Lajos Molnár
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  2. Patrícia Szokol
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Corresponding author

Correspondence to Lajos Molnár.

Additional information

The authors were supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences. The research was partially supported also by the European Union and the European Social Fund through project Supercomputer, the national virtual lab (grant no.: TAMOP-4.2.2.C-11/1/KONV-2012-0010).

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Molnár, L., Szokol, P. Transformations Preserving Norms of Means of Positive Operators and Nonnegative Functions. Integr. Equ. Oper. Theory 83, 271–290 (2015). https://doi.org/10.1007/s00020-015-2241-6

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  • DOI: https://doi.org/10.1007/s00020-015-2241-6

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