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Math Help - Isomorphisms

  1. #1
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    Isomorphisms

    Hello,

    i try to understand a few isomorphisms, which i have found without comment in a text:
    Let H be a Hilbert space and T \in B(H). Then we have the following identities:

     <br />
1)ker(T)^\bot = \overline{imT^*}
     <br />
2)(kerT^* )^\bot = \overline{imT}<br />
    I know that we can decompose the Space H by H=ker(T)\oplus ker(T)^\bot
    And by the isomorphism above (1) we get H=ker(T)\oplus \overline{imT^*}
    Analogue we have also a decomposition H=\overline{ImT}\oplus imT^\bot =

    But then the text says:
     <br />
3) \overline{imT^*}=\overline{imT} \n<br />
4) ker T = ker T^*<br />

    Are these equations always true? I mean T is bounded in this case, but isn't it necessary that T is injective or something like that? It would be nice, if someone can explain those eqations to me.

    Regards
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Isomorphisms

    Quote Originally Posted by Sogan View Post
    Hello,

    i try to understand a few isomorphisms, which i have found without comment in a text:
    Let H be a Hilbert space and T \in B(H). Then we have the following identities:

     <br />
1)ker(T)^\bot = \overline{imT^*}
     <br />
2)(kerT^* )^\bot = \overline{imT}<br />
    I know that we can decompose the Space H by H=ker(T)\oplus ker(T)^\bot
    And by the isomorphism above (1) we get H=ker(T)\oplus \overline{imT^*}
    Analogue we have also a decomposition H=\overline{ImT}\oplus imT^\bot =

    But then the text says:
     <br />
3) \overline{imT^*}=\overline{imT} \n<br />
4) ker T = ker T^*<br />

    Are these equations always true? I mean T is bounded in this case, but isn't it necessary that T is injective or something like that? It would be nice, if someone can explain those eqations to me.

    Regards
    Well, it's definitely true if T is normal, right? So why don't you take a typical non-normal [tex]T[tex] and try it out?
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  3. #3
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    Re: Isomorphisms

    Ah ok, thank you i will check it out today. But are the first two eqations also true just for normal operators?

    Thanks!
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Isomorphisms

    Quote Originally Posted by Sogan View Post
    Ah ok, thank you i will check it out today. But are the first two eqations also true just for normal operators?

    Thanks!
    By the previous equations doesn't the second imply the first (remember kernels are closed of course)?
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