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Math Help - T is a normal operator; prove that R(T) = R(T*)

  1. #1
    Member Last_Singularity's Avatar
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    T is a normal operator; prove that R(T) = R(T*)

    Question: Let T is a normal operator on a finite-dimensional inner product space V. Show that N(T)=N(T^*) and that R(T)=R(T^*).

    I think that I got the first part. Suppose x \in N(T) Then T(x)=0. But ||T(x)|| = ||T^*(x)|| for all x so T^*(x)=0 as well. So x \in N(T^*), which means that N(T) \subseteq N(T^*). The same logic goes the other way around and we conclude N(T) = N(T^*).

    But what about R(T)=R(T^*)? I know that R(T^*) = N(T)^{perp} but how does that help?
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  2. #2
    Senior Member Shanks's Avatar
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    R(T^*) = N(T)^{perp}=N(T^*)^{perp}=R(T)
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