Advertisement

Product formulas in the framework of mean ergodic theorems

  • J. Z. BernádEmail author
Original Paper

Abstract

An extension of Chernoff’s product formula for one-parameter functions taking values in the space of bounded linear operators on a Banach space is given. Essentially, the n-th one-parameter function in the product formula is mapped by the n-th iterate of a contraction acting on the space of the one-parameter functions. The motivation to study this specific product formula lies in the growing field of dynamical control of quantum systems, involving the procedure of dynamical decoupling and also the Quantum Zeno effect.

Keywords

Chernoff’s product formula Mean ergodic theorems Strongly continuous semigroups 

Mathematics Subject Classification

47D03 47A35 47N50 

Notes

Acknowledgements

This work is supported by the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 732894 (FET Proactive HOT).

References

  1. 1.
    Arendt, W., Ulm, M.: Trotter’s product formula for projections. Ulmer Seminare, pp. 394–399 (1997)Google Scholar
  2. 2.
    Arenz, C., Hillier, R., Fraas, M., Burgarth, D.: Distinguishing decoherence from alternative quantum theories by dynamical decoupling. Phys. Rev. A 92(2), 022102 (2015)CrossRefGoogle Scholar
  3. 3.
    Barankai, N., Zimborás, Z.: Generalized quantum Zeno dynamics and ergodic means. arXiv:1811.02509
  4. 4.
    Bernád, J,Z.: Dynamical control of quantum systems in the context of mean ergodic theorems. J. Phys. A Math. Theor. 50(6), 065303 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2(2), 238–242 (1968)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chernoff, P.R.: Semigroup product formulas and addition of unbounded operators. Bull. Am. Math. Soc. 76(2), 395–398 (1970)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. Interscience Publishers, New York (1964)zbMATHGoogle Scholar
  8. 8.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)zbMATHGoogle Scholar
  9. 9.
    Exner, P., Ichinose, T.: A product formula related to quantum zeno dynamics. Ann. H. Poincaré 6(6), 195–215 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Facchi, P., Pascazio, S.: Quantum Zeno dynamics: mathematical and physical aspects. J. Phys. A Math. Theor. 4149, 493001 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kato, T.: Remarks on pseudo-resolvents and infinitesimal generators of semi-groups. Proc. Jpn. Acad. 35(8), 467–468 (1959)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kato, T., Tanabe, H.: On the abstract evolution equation. Osaka J. Math. 14(1), 107–133 (1962)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Krengel, U.: Ergodic Theorems. de Gruyter, Berlin (1985)CrossRefGoogle Scholar
  14. 14.
    Lorch, E.R.: Means of iterated transformations in reflexive vector spaces. Bull. Am. Math. Soc. 45(1), 945–947 (1939)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Matolcsi, M., Shvidkoy, R.: Trotter’s product formula for projections. Arch. Math. 81(3), 309–317 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Möbus, T., Wolf, M.M.: Quantum Zeno effect generalized. J. Math. Phys. 60(5), 052201 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Trotter, H.: Approximation of semi-groups of operators. Pac. J. Math. 8(4), 887–919 (1958)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Trotter, H.: On the product of semigroups of operators. Proc. Am. Math. Soc. 10(4), 545–551 (1959)CrossRefGoogle Scholar
  19. 19.
    Vuillermot, P.-A.: A generalization of Chernoff’s product formula for time-dependent ope-rators. J. Funct. Anal. 259(11), 2923–2938 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Vuillermot, P.-A., Wreszinski, W.F., Zagrebnov, V.A.: A general Trotter-Kato formula for a class of evolution operators. J. Funct. Anal. 257(7), 2246–2290 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Yosida, K.: Mean ergodic theorem in Banach spaces. Proc. Imp. Acad. Tokyo 14(8), 292–294 (1938)MathSciNetCrossRefGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of MaltaMsidaMalta

Personalised recommendations