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

 

Path connectivity of idempotents on a Hilbert space


Authors: Yan-Ni Chen, Hong-Ke Du and Hai-Yan Zhang
Journal: Proc. Amer. Math. Soc. 136 (2008), 3483-3492
MSC (2000): Primary 47A05, 46C07, 15A09
Published electronically: May 30, 2008
MathSciNet review: 2415032
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Abstract: Let $ P$ and $ Q$ be two idempotents on a Hilbert space. In 2005, J. Giol in [Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229(2005) 405-423] had established that, if $ P+Q-I$ is invertible, then $ P$ and $ Q$ are homotopic with $ \tilde{s}(P,Q)\leq 2.$ In this paper, we have given a necessary and sufficient condition that $ \tilde{s}(P,Q)\leq 2,$ where $ \tilde{s}(P,Q)$ denotes the minimal number of segments required to connect not only from $ P$ to $ Q$, but also from $ Q$ to $ P$ in the set of idempotents.


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Additional Information

Yan-Ni Chen
Affiliation: Department of Mathematics, Shaanxi University of Technology, Hanzhong 723001, People’s Republic of China
Email: operatorguy@126.com

Hong-Ke Du
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
Email: hkdu@snnu.edu.cn

Hai-Yan Zhang
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09194-6
PII: S 0002-9939(08)09194-6
Keywords: Idempotent, orthogonal projection, homotopic, path connectivity
Received by editor(s): July 18, 2006
Received by editor(s) in revised form: April 11, 2007
Published electronically: May 30, 2008
Additional Notes: This research was partially supported by the National Natural Science Foundation of China (10571113)
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2008 American Mathematical Society