Decompose Hilbert space using bounded linear operators

Question

if $A$ is a bounded linear operator on $H$, can we decompose $H$ by $ H=\overline{\operatorname{ran} A} \oplus \operatorname{ker} A $? It seem to me that we need to show that $A$ is actually a closed operator. If that's not right, is there a possible way to decompose $H$ using range and kernal of $A$? Any help is appreciated.

Answer 1

First of all, a bounded operator is obviously closed.

Secondly, we have $H=(\ker A)^\perp \oplus \ker A$ and $(\ker A)^\perp=\overline{\text{ran}\, A^*}$. In particular, if $A^*=A$ (self-adjoint) then $$H=\overline{\text{ran}\, A} \oplus \ker A.$$