# Closedness of the Rang of the Product of Projections in Hilbert Modules

Volume 2, Issue 4, pp 588--593
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### Authors

Kamran Sharifi - Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran

### Abstract

Suppose P and Q are orthogonal projections between Hilbert C*-modules, then PQ has closed range if and only if Ker(P)+Ran(Q) is an orthogonal summand, Ker(Q)+ Ran(P) is an orthogonal summand.

### Share and Cite

##### ISRP Style

Kamran Sharifi, Closedness of the Rang of the Product of Projections in Hilbert Modules, Journal of Mathematics and Computer Science, 2 (2011), no. 4, 588--593

##### AMA Style

Sharifi Kamran, Closedness of the Rang of the Product of Projections in Hilbert Modules. J Math Comput SCI-JM. (2011); 2(4):588--593

##### Chicago/Turabian Style

Sharifi, Kamran. "Closedness of the Rang of the Product of Projections in Hilbert Modules." Journal of Mathematics and Computer Science, 2, no. 4 (2011): 588--593

### Keywords

• Hilbert C*-module
• projection
• Moore-Penrose inverse
• closed range.

•  46L08
•  47A05
•  15A09
•  46L05

### References

• [1] R. Bouldin, The product of operators with closed range, Tôhoku Math. J., 25 (1973), 359--363

• [2] R. Bouldin, Closed range and relative regularity for products, J. Math. Anal. Appl., 61 (1977), 397--403

• [3] F. Deutsch, The angle between subspaces in Hilbert space, in: Approximation theory, wavelets and applications, 1995 (1995), 107--130

• [4] M. Frank, K. Sharifi, Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem, J. Operator Theory, 63 (2010), 271--282

• [5] M. Frank, K. Sharifi , Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C*-modules , J. Operator Theory, Vol. 64, 377--386 (2008)

• [6] J. M. Gracia-Bonda, J. C. Várilly, H. Figueroa, Elements of noncommutative geometry, Birkhäuser, Boston (2000)

• [7] S. Izumino, The product of operators with closed range and an extension of the revers order law, Tôhoku Math. J., 34 (1982), 43--52

• [8] J. J. Koliha, The Drazin and Moore-Penrose inverse in C*-algebras, Proc. Roy. Irish Acad. Sect. A, 99 (1999), 17--27

• [9] J. J. Koliha, V. Rakočević, Fredholm properties of the difference of orthogonal projections in a Hilbert space, Integr. Equ. Oper. Theory , 52 (2005), 125--134

• [10] E. C. Lance , Hilbert C*-modules: a toolkit for operator algebraists, Cambridge University Press, Cambridge (1995)

• [11] G. J. Murphy, C*-algebras and Operator Theory, Academic Press, New York (1990)

• [12] I. Raeburn, D. P. Williams, Morita Equivalence and Continuous Trace C*-algebras, American Mathematical Soc., Providence (1998)

• [13] K. Sharifi, Descriptions of partial isometries on Hilbert C*-modules, Linear Algebra Appl., 431 (2009), 883--887

• [14] K. Sharifi , Groetsch’s Representation of Moore-Penrose inverses and illposed problems in Hilbert C*-modules, J. Math. Anal. Appl., 365 (2010), 646--652

• [15] Q. Xu, L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C*-modules, Linear Algebra Appl., 428 (2008), 992--1000