On the Generality of Orthogonal Projections and e-Projections in Banach Algebras

DOI: 10.4236/apm.2012.25044   PDF   HTML     3,466 Downloads   6,046 Views  

Abstract

In this paper we develop the orthogonal projections and e-projections in Banach algebras. We prove some necessary and sufficient conditions for them and their spectrums. We also show that the sum of two generalized orthogonal projections u and v is a generalized orthogonal projection if, uv=vu=0. Our results generalize the results obtained for bounded linear operators on Hilbert spaces.

Share and Cite:

M. Asgari, S. Karimizad and H. Rahimi, "On the Generality of Orthogonal Projections and e-Projections in Banach Algebras," Advances in Pure Mathematics, Vol. 2 No. 5, 2012, pp. 318-322. doi: 10.4236/apm.2012.25044.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] E. Berkson, “Hermitian Projections and Orthogonality in Banach Spaces,” Proceedings London Mathematical Society, Vol. 3, No. 24, 1972, pp. 101-118. doi:10.1112/plms/s3-24.1.101
[2] C. Schmoeger, “Generalized Projections in Banach Algebras,” Linear Algebra and its Applications, Vol. 430, No. 10, 2009, pp. 601-608. doi:10.1016/j.laa.2008.07.020
[3] H. Du and Y. Li, “The Spectral Characterization of Generalized Projections,” Linear Algebra and its Applications, Vol. 400, 2005, pp. 313-318. doi:10.1016/j.laa.2004.11.027
[4] I. Groβ and G. Trenkler, “Generalized and Hyper Generalized Projectors,” Linear Algebra and its Applications, Vol. 264, 1997, pp. 463-474.
[5] L. Lebtahi and N. thome, “A Note on κ-Generalized Projection,” Linear Algebra and its Applications, Vol. 420, 2007, pp. 572-575. doi:10.1016/j.laa.2006.08.011
[6] F. F. Bonsal and J. Duncan, “Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras,” Cambridge University Press, Cambridge, 1971.
[7] W. Rudin, “A Course in Functional Analysis,” McGraw Hill, New York, 1973.
[8] I. S. Murphy, “A Note on Hermitian elements of a Banach Algebra,” Journal London Mathematical Society, Vol. 3, No. 6, 1973, pp. 427-428. doi:10.1112/jlms/s2-6.3.427
[9] J. B. Conway, “A Course in Functional Analysis,” Springer-Verlag Inc., New York, 1985.

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.