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Math Help - Adjoint Operator

  1. #1
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    Unhappy Adjoint Operator

    Hi!!!
    I don't know how to solve this.
    Let V a inner product space, T: V \rightarrow V a linear aplication and T^* its adjoint.
    Prove that ker(TT^*+T^*T) = ker(T)\cap ker(T^*)
    Obviously there is an easy inclusion
    Thanks!!
    Everk
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  2. #2
    Super Member girdav's Avatar
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    Let x\in V such that T^*Tx+TT^*x =0. We have 0\leq \langle Tx,Tx\rangle =\langle T^*Tx,x\rangle = -\langle TT^*x,x\rangle =-\langle T^*x,T^*x\rangle\leq 0.
    Last edited by girdav; May 24th 2011 at 09:05 AM. Reason: typo
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