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Math Help - Adjoint of Bounded Operators

  1. #1
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    Adjoint of Bounded Operators

    If S,T:H\rightarrow H are bounded operators, show that (ST)^*=T^*S^*.

    I'm assuming that H is a Hilbert space, although it doesn't say this in the question. I'm really not sure where to start with this. All I have is that since S,T are bounded there adjoints S^* and T^* exist and that:

    <Tx,y>=<x,T^*y> for all x\in H, y\in H
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  2. #2
    Member Focus's Avatar
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    Start by applying the adjoint principal to Tx, i.e.
    <br />
\langle S(Tx),y \rangle=\langle Tx,S^*y \rangle<br />

    I hope you can now see the next step.
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  3. #3
    Member mabruka's Avatar
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    This kind of identities are usually proved as follows:

    For any x,y in the hilbert space
    \langle x,(ST)^* y \rangle=\langle STx,y\rangle =\langle Tx,S^*y\rangle=\langle x,T^*S^*y\rangle


    From where

    \langle x,(ST)^* y \rangle -\langle x,T^*S^*y\rangle  =  0 for all  x,y \in \mathcal H

    Therefore (ST)^*=T^*S^*

    Here it is used that if \langle x,(B-C)y \rangle = 0 for all x,y \in \mathcal H then B=C
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