Glasnik Matematicki, Vol. 45, No.2 (2010), 505-512.

CONTINUITY OF THE POLAR DECOMPOSITION FOR UNBOUNDED OPERATORS ON HILBERT C*-MODULES

Kamran Sharifi

Abstract.   For unbounded operators t,s between Hilbert C*-modules which admit the polar decompositions V|t|, W|s|, respectively, we obtain an explicit upper bound estimate for the gap between t and s in terms of the norm of the bounded operators V-W, C_|t|-C_|s| and C_|t^*|-C_|s^*|, where C_|t| and C_|s| are the Cayley transforms of |t| and |s|. The result are used to drive a criterion for continuity of the polar decomposition for unbounded operators between Hilbert C*-modules.

2000 Mathematics Subject Classification.   46L08, 46C50, 47C15, 47B50.

Key words and phrases.   Hilbert C*-module, unbounded operator, gap metric, Cayley transform.

DOI: 10.3336/gm.45.2.15

References:

1. W. Arveson, An Invitation to C*-algebras, Springer, New York, 1976.
   MathSciNet    

2. S. Baaj and P. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les C*-modules hilbertiens, C. R. Acad. Sc., Paris, Series I 296 (1983), 875-878.
   MathSciNet    

3. N. Castro-González, J. J. Koliha and V. Rakočević, Continuity and general perturbation of the Drazin inverse for closed linear operators, Abstr. Appl. Anal. 7 (2002), 335-347.
   MathSciNet     CrossRef

4. M. Frank, Geometrical aspects of Hilbert C*-modules, Positivity 3 (1999), 215-243.
   MathSciNet     CrossRef
5. M. Frank and K. Sharifi, Adjointability of densely defined closed operators and the Magajna-Schweizer theorem, J. Operator Theory 63 (2010), 271-282.
   MathSciNet    

6. M. Frank and K. Sharifi, Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C*-modules, J. Operator Theory 64 (2010), 377-386.
   MathSciNet    

7. B. Guljaš, Unbounded operators on Hilbert C*-modules over C*-algebras of compact operators, J. Operator Theory 59 (2008), no. 1, 179-192.
   MathSciNet    

8. T. Kato, Perturbation theory for linear operators, Springer Verlag, New York, 1984.
   MathSciNet    

9. J. J. Koliha and V. Rakočević, Continuity of the Drazin inverse II, Studia Math. 131 (1998), no. 2, 167-177.
   MathSciNet    
10. E. C. Lance, Hilbert C*-modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995.
    MathSciNet    

11. B. Magajna, Hilbert C*-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125(3) (1997), 849-852.
    MathSciNet     CrossRef

12. K. Sharifi, The gap between unbounded regular operators, to appear in J. Operator Theory.

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

14. K. Sharifi, Topological approach to unbounded operators on Hilbert C*-modules, to appear in Rocky Mountain J. Math.

15. N. E. Wegge-Olsen, K-theory and C*-algebras: a Friendly Approach, Oxford University Press, Oxford, England, 1993.
    MathSciNet    

16. S. L. Woronowicz, Unbounded elements affiliated with C*-algebras and noncompact quantum groups, Comm. Math. Phys. 136 (1991), 399-432.
    MathSciNet     CrossRef