Let $A$ be a bounded operator on a Hilbert space $H$ with two invariant subspaces $M$ and $N$ s.t. $N \subset M$, dim$(M \cap N^{\perp})> 1$, and have no invariant subspaces between $N$ and $M$. Then, show that, there exists an operator $B$ on $H$ which has no proper invariant subspace.
All I want a hint for constructing $B$ with the help of $A$ and given conditions, even a little hint will be appreciated. Thanks in advance.
The hint is to think about the invariant subspaces of $A$ the lie in $M\cap N^\perp$.