Glasnik Matematicki, Vol. 46, No. 2 (2011), 433-438.

AN IMPLICIT DIVISION OF BOUNDED AND UNBOUNDED LINEAR OPERATORS WHICH PRESERVES THEIR PROPERTIES

Mohammed Hichem Mortad

Abstract.   We give an answer to the following problem: Given two linear operators A and B such that BA and A verify some property P, then when does B verify the same property P? Of course, we have to assume that B satisfies some condition Q independent of (or weaker than) P. This problem is solved in the setting of both bounded and unbounded operators on a Hilbert space. Some interesting counterexamples are also given.

2000 Mathematics Subject Classification.   47A05.

Key words and phrases.   Products of operators, bounded and unbounded operators, self-adjoint, closed and normal operators.

DOI: 10.3336/gm.46.2.12

References:

1. E. Albrecht and P. G. Spain, When products of selfadjoints are normal, Proc. Amer. Math. Soc. 128 (2000), 2509-2511.
   MathSciNet     CrossRef

2. J. B. Conway, A course in functional analysis, Springer, New York, 1990.
   MathSciNet    

3. M. R. Embry, Similarities involving normal operators on Hilbert space, Pacif. J. Math. 35 (1970), 331-336.
   MathSciNet     CrossRef

4. B. Fuglede, A commutativity theorem for normal operators, Proc. Nati. Acad. Sci. 36 (1950), 35-40.
   MathSciNet    

5. T. Furuta, Invitation to linear operators. From matrices to bounded linear operators on a Hilbert space, Taylor & Francis, Ltd., London, 2001.
   MathSciNet    

6. M. H. Mortad, An All-Unbounded-Operator Version of the Fuglede-Putnam Theorem, Complex Anal. Oper. Theory, to appear.
   CrossRef

7. M. H. Mortad, An application of the Putnam-Fuglede theorem to normal products of self-adjoint operators, Proc. Amer. Math. Soc. 131 (2003), 3135-3141.
   MathSciNet     CrossRef

8. M. H. Mortad, On some product of two unbounded self-adjoint operators, Integral Equations Operator Theory 64 (2009), 399-407.
   MathSciNet     CrossRef

9. M. H. Mortad, Similarities involving unbounded normal operators, Tsukuba J. Math. 34 (2010), 129-136.
   MathSciNet    

10. W. Rehder, On the product of self-adjoint operators, Internat. J. Math. and Math. Sci. 5 (1982), 813-816.
    MathSciNet     CrossRef

11. W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991 (2nd edition).
    MathSciNet