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Math Help - Hilbert Adjoint Operator T^{\ast}

  1. #1
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    Hilbert Adjoint Operator T^{\ast}

    (from Kreyszig p.200 sec. 3.9 Hilbert Adjoint Operators)
    -------------------------------------------------

    Let H_1 and H_2 be Hilbert Spaces and
    T: H_1 \longrightarrow H_2 a bounded linear operator.

    If M_1 \subset H_1 and M_2 \subset H_2 are such that

    T(M_1) \subset M_2, show that

    M^{\perp}_1 \supset T^{\ast}(M_2^{\perp}).
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  2. #2
    Super Member Rebesques's Avatar
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    Just note that \langle M_1,T^* (M_2^{\perp})\rangle=\langle T(M_1),M_2^{\perp}\rangle.
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