Let $H_1,H_2 $ and $H$ be Hilbert spaces. Let $A:H_1\rightarrow H$ and $B:H_2\rightarrow H $ be two bounded linear operators with $N(A^*)\cap N(B^*)\neq \{0\}.$ How can we prove that $\overline{\overline{R(A)}+\overline {R(B)}}\neq H$ ?
Here, $N(A^*)\neq \{0\}$ and $N(B^*)\neq \{0\}$, which means $\overline{R(A)}\neq H$ and $\overline{R(B)}\neq H$. I am unable to proceed further. Kindly give some hints.
It seems like you're using the fact that $N(A^\star)^\perp = \overline{R(A)}$ - see here.
Another fact you might like to use is that if $L_1$, $L_2$ are two closed subspaces of a Hilbert space $H$, then $(L_1 \cap L_2)^\perp = \overline{L_1^\perp + L_2^\perp}$ - see here.
These two facts together should be enough to solve the problem.