Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-08-09194-6
Published electronically: May 30, 2008
MathSciNet review: 2415032
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Y. N. Chen, H. K. Du, Idempotency of linear combinations of two idempotents on a Hilbert space, Acta. Math. Sinica 50 (2007) 1171-1176. MR 2370349
  • 2. Y. N. Chen, H. K. Du, Y. F. Pang, A simplification of the Kovarik formula, Journal of Mathematical Analysis and its Applications 331 (2007) 13-20. MR 2305984
  • 3. C. Y. Deng, H. K. Du, Common complements of two subspaces and an answer to Groß's question, Acta. Math. Sinica 49 (2006) 1099-1112.MR 2285414 (2008b:47003)
  • 4. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Springer-Verlag, New York, 2003. MR 1634900 (99c:47001)
  • 5. H. K. Du, C. Y. Deng, The representation and characterization of Drazin inverses of operators on a Hilbert space, Linear Algebra Appl. 407 (2005) 117-124. MR 2161918 (2006d:47001)
  • 6. H. K. Du, X. Y. Yao, C. Y. Deng, Invertiblity of linear combinations of two idempotents, Proc. Amer. Math. Soc. 134 (2006) 1451-1457. MR 2199192 (2006k:47004)
  • 7. H. K. Du, W. F. Wang, Y. T. Duan, Path connectivity of $ k$-generalized projectors, Linear Algebra Appl. 422 (2007) 712-720. MR 2305151 (2008b:47004)
  • 8. J. Esterle, Polynomial connections between projections in Banach algebras. Bull. London Math. Soc. 15 (1983) 253-254. MR 697127 (84g:46069)
  • 9. J. Esterle, J. Giol, Polynomial and polygonal connections between idempotents in finite-dimensional real algebras, Bull. London Math. Soc. 36 (2004) 378-382. MR 2038725 (2005b:46100)
  • 10. L. R. Fillmore, J. P. Williams, On operator ranges, Advances in Math. 7 (1971) 254-281. MR 0293441 (45:2518)
  • 11. J. Giol, Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229 (2005) 405-423. MR 2182594 (2006h:47062)
  • 12. J. Zemánek, Idempotents in Banach algebra, Bull. London Math. Soc. 11 (1979) 177-183.MR 0541972 (80h:46073)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A05, 46C07, 15A09

Retrieve articles in all journals with MSC (2000): 47A05, 46C07, 15A09


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: https://doi.org/10.1090/S0002-9939-08-09194-6
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

American Mathematical Society