DOI QR코드

DOI QR Code

A REMARK ON ERGODICITY OF QUANTUM MARKOVIAN SEMIGROUPS

Ko, Chul-Ki

  • Published : 2009.01.31

Abstract

The aim of this paper is to find the set of the fixed elements and the set of elements for which equality holds in Schwarz inequality for the KMS-symmetric Markovian semigroup $\{S_t\}_{t{\geq}0}$ given in [10]. As an application, we study some properties such as the ergodicity and the asymptotic behavior of the semigroup.

Keywords

quantum Markovian semigroups;ergodicity

References

  1. O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics, Springer-Verlag, New York-Heidelberg-Berlin, vol I (1979)
  2. F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von Neu-mann algebras, J. Funct. Anal. 147 (1997), 259-300. https://doi.org/10.1006/jfan.1996.3063
  3. F. Cipriani, F. Fagnola, and J. M. Lindsay, Spectral analysis and Feller properties for quantum Ornstein-Uhlenbeck semigroups, Comm. Math. Phys. 210 (2000), 85-105. https://doi.org/10.1007/s002200050773
  4. F. Fagnola and R. Quezada, Two-photon and emission process, Inf. Dim. Anal. Quantum Probab. Related Topics 8 (2005), 573-591. https://doi.org/10.1142/S0219025705002116
  5. F. Fagnola and R. Rebolledo, The approach to equilibrium of a class of quantum dynamical semigroups, Inf. Dim. Anal. Quantum Probab. Related Topics 1 (1998), 561-572. https://doi.org/10.1142/S0219025798000302
  6. F. Fagnola and R. Rebolledo, Notes on the qualitative behaviour of quantum Markov semigroups, Lecture Notes in Math. Open quantum systems. III, 1882, Springer, Berlin, (2006), 161-205.
  7. A. Frigerio, Stationary states of quantum dynamical semigroups, Comm. Math. Phys. 63 (1978), 269-276. https://doi.org/10.1007/BF01196936
  8. A. W. Majewski and B. Zegarlinski, Quantum stochastic dynamics I, Spin systems on a lattice, MPEJ 1 (1995) Paper 2.
  9. A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003), 189-209. https://doi.org/10.1016/S0022-1236(02)00151-9
  10. Y. M. Park, Remarks on the structure of Dirichlet forms on standard forms of von Neumann algebras, Inf. Dim. Anal. Quantum Probab. Related Topics 8 (2005), 179-197. https://doi.org/10.1142/S0219025705001925
  11. Y. M. Park, Ergodic property of Markovian semigroups on standard forms of von Neumann algebras, J. Math. Phys. 46 (2005), 113507-1-113507-13. https://doi.org/10.1063/1.2113067
  12. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992.
  13. O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics, Springer-Verlag, New York-Heidelberg-Berlin, vol. II (1981).