1. ## 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:

$
1)ker(T)^\bot = \overline{imT^*}$

$
2)(kerT^* )^\bot = \overline{imT}
$

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:
$
3) \overline{imT^*}=\overline{imT} \n
4) ker T = ker T^*
$

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

2. ## Re: Isomorphisms

Originally Posted by Sogan
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:

$
1)ker(T)^\bot = \overline{imT^*}$

$
2)(kerT^* )^\bot = \overline{imT}
$

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:
$
3) \overline{imT^*}=\overline{imT} \n
4) ker T = ker T^*
$

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?

3. ## 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!

4. ## Re: Isomorphisms

Originally Posted by Sogan
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)?