QUANTUM DYNAMICAL SEMIGROUP AND ITS ASYMPTOTIC BEHAVIORS

Title & Authors
QUANTUM DYNAMICAL SEMIGROUP AND ITS ASYMPTOTIC BEHAVIORS
Choi, Veni;

Abstract
In this study we consider quantum dynamical semi-group with a normal faithful invariant state. A quantum dynamical semigroup $\small{\alpha\;=\;\{{\alpha}_t\}_{t{\geq}0}}$ is a class of linear normal identity-preserving mappings on a von Neumann algebra M with semigroup property and some positivity condition. We investigate the asymptotic behaviors of the semigroup such as ergodicity or mixing properties in terms of their eigenvalues under the assumption that the semigroup satisfies positivity. This extends the result of [13] which is obtained under the assumption that the semi group satisfy 2-positivity.
Keywords
quantum dynamical semigroup;positivity;Schwarz inequality;Jordan product;ergodicity;weak mixing;
Language
English
Cited by
1.
A Survey on Invariance and Ergodicity of Quantum Markov Semigroups, Stochastic Analysis and Applications, 2014, 32, 3, 480
2.
Recurrence and Transience of Quantum Markov Semigroups, Stochastic Analysis and Applications, 2015, 33, 1, 123
References
1.
Operator algebras and quantum statistical mechanics I., 1979.

2.
One-parameter semigroups, 1980.

3.
Comm. Math. Phys., 1977. vol.54. 3, pp.293-297

4.
Math Z., 1982. vol.180. 2, pp.275-286

5.
Lecture Notes in Math., 1986. vol.1184. pp.369-425

6.
Math. Proc. Cambridge Philos. Soc., 1992. vol.111. 1, pp.181-192

7.
J. Math. Anal. Appl., 1998. vol.221. 1, pp.13-32

8.
Math. Z., 2000. vol.235. 3, pp.615-626

9.
Harmonic analysis of operators on hilbert space, 1970.

10.
Comm. Math. Phys., 1982. vol.85. 1, pp.129-142

11.
Theory of operator algebras I., 1979.

12.
Studia Math., 1985. vol.81. 3, pp.285-291

13.
J. Math. Anal. Appl., 1982. vol.86. 2, pp.411-424