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 $\displaystyle T \in B(H). $Then we have the following identities:

$\displaystyle

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

$\displaystyle

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

$

I know that we can decompose the Space H by $\displaystyle H=ker(T)\oplus ker(T)^\bot$

And by the isomorphism above (1) we get $\displaystyle H=ker(T)\oplus \overline{imT^*}$

Analogue we have also a decomposition $\displaystyle H=\overline{ImT}\oplus imT^\bot = $

But then the text says:

$\displaystyle

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