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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On residualities in the set of Markov operators on $\mathcal{C}_1$


Authors: Wojciech Bartoszek and Beata Kuna
Journal: Proc. Amer. Math. Soc. 133 (2005), 2119-2129
MSC (2000): Primary 46L55, 47A35; Secondary 37A55, 47B60
Published electronically: February 15, 2005
MathSciNet review: 2137879
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Abstract: We show that the set of those Markov operators on the Schatten class $\mathcal{C}_1$ such that $\lim_{n \to \infty} \Vert P^n - Q \Vert = 0$, where $Q$ is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators $P$ form a norm dense $G_{\delta}$. Surprisingly, for the strong operator topology operators the situation is quite the opposite.


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Additional Information

Wojciech Bartoszek
Affiliation: Department of Mathematics, Gdańsk University of Technology, ul. Narutowicza 11/12, 80 952 Gdańsk, Poland

Beata Kuna
Affiliation: Department of Mathematics, Gdańsk University of Technology, ul. Narutowicza 11/12, 80 952 Gdańsk, Poland

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07776-2
PII: S 0002-9939(05)07776-2
Keywords: Schatten classes, Markov operator, mixing operator, residuality
Received by editor(s): May 16, 2003
Received by editor(s) in revised form: April 7, 2004
Published electronically: February 15, 2005
Additional Notes: The authors thank the referee for pointing out a gap in the first version of the proof of Theorem 2.6
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.