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Math Help - Adjoint Operation

  1. #1
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    Adjoint Operation

    Let H be a Hilbert Space and T be an operator in H.
    Prove that the mapping T\rightarrow T^* is an one to one and onto mapping of B(H) into itself,where T^* is the adjoint of T and B(H) is the set of bounded operator on H.

    Since we know that T uniquely determines the adjoint T^* and by the fact that T^* is an operator in H,hence,we can claim that the mapping from T\rightarrow T^* is one to one and onto.

    Is this the right way to prove?Can anyone help?
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  2. #2
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    What you need is not that "T uniquely determines T*" but that "T* uniquely determines T". Otherwise it is possible that two different Ts determine the same T* and the mapping would not be one to one.
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  3. #3
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    HallsofIvy,can you give me some hints in proving the one to one?
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  4. #4
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    Use the definition of "adjoint"! If T* is the adjoint of T, the <Tu, v>= <u, T*v> for all u and v in H. < , > is the inner product in H. Suppose that, for a given T, there were two adjoints, T_1* and T_*. Then we must have [tex]<Tu, v>= <u, T_1*v>[tex] and <Tu, v>= <u, T_2*v> so that <u, T_1*v>= <u, T_2*v> for all u and v in H. Can you show that T_1*= T_2* from that?
    Last edited by HallsofIvy; October 11th 2009 at 09:49 AM.
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