Preserver Problems Related to Quasi-Arithmetic Means of Invertible Positive Operators

Article
  • 37 Downloads

Abstract

In this paper we mainly discuss different preserver problems on the cone of positive definite matrices which are related to certain quasi-arithmetic means.

Keywords

Positive definite matrices Quasi-arithmetic means Nonlinear preservers Geodesics 

Mathematics Subject Classification

Primary: 15B48 47A64 47B49 Secondary: 53C22 

Notes

Acknowledgements

The authors were supported by the National Research, Development and Innovation Office NKFIH Reg. No. K115383.

References

  1. 1.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefMATHGoogle Scholar
  2. 2.
    Daróczy, Z.: Gauss-composition of means and the solution of the Matkowski–Sutô problem. Publ. Math. Debr. 61, 157–218 (2002)MATHGoogle Scholar
  3. 3.
    Dixmier, J.: Von Neumann Algebras. North-Holland Publishing Company, Amsterdam (1981)MATHGoogle Scholar
  4. 4.
    Fujii, J.I.: Path of quasi-means as a geodesic. Linear Algebra Appl. 434, 542–558 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gaál, M., Nagy, G.: Maps on positive operators preserving Rényi type relative entropies and maximal f-divergences. Lett. Math. Phys. 108(2), 425–443 (2018)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hiai, F., Petz, D.: Riemannian metrics on positive definite matrices related to means. Linear Algebra Appl. 430, 3105–3130 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hill, R.D.: Linear transformations which preserve Hermitian matrices. Linear Algebra Appl. 6, 257–262 (1973)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kadison, R.V.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math. 56, 494–503 (1952)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Molnár, L.: Maps on the positive definite cone of a \(C^*\)-algebra preserving certain quasi-entropies. J. Math. Anal. Appl. 447, 206–221 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Molnár, L.: Order automorphisms on positive definite operators and a few applications. Linear Algebra Appl. 434, 2158–2169 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Molnár, L.: Maps preserving general means of positive operators. Electron. J. Linear Algebra 22, 864–874 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Molnár, L.: Maps preserving the harmonic mean or the parallel sum of positive operators. Linear Algebra Appl. 430, 3058–3065 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Molnár, L.: Maps preserving the geometric mean of positive operators. Proc. Am. Math. Soc. 137, 1763–1770 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Molnár, L.: Order-automorphisms of the set of bounded observables. J. Math. Phys. 42, 5904–5909 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Molnár, L., Szokol, P.: Transformations preserving norms of means of positive operators and nonnegative functions. Integral Equ. Oper. Theory 83, 271–290 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nagy, G.: Preservers for the \(p\)-norm of linear combinations of positive operators. Abstr. Appl. Anal. 2014, 434121-1–434121-9 (2014)Google Scholar
  18. 18.
    Palmer, T.W.: Banach Algebras and the General Theory of *-Algebras: Volume 1, Encyclopedia of Mathematics and Its Applications, vol. 49. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  19. 19.
    Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, Boston (1990)MATHGoogle Scholar
  20. 20.
    Szokol, P., Tsai, M.C., Zhang, J.: Preserving problems of geodesic-affine maps and related topics on positive definite matrices. Linear Algebra Appl. 483, 293–308 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Functional Analysis Research GroupUniversity of SzegedSzegedHungary
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary

Personalised recommendations