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

Affiliation(s)

Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran.

Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran.

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.

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

KEYWORDS

Generalized Orthogonal Projection; Orthogonal Projection; Generalized e-Projection; e-Projection

Generalized Orthogonal Projection; Orthogonal Projection; Generalized e-Projection; e-Projection

Cite this paper

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.

M. Asgari, S. Karimizad and H. Rahimi, "On the Generality of Orthogonal Projections and e-Projections in Banach Algebras,"

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.

[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.