# There exists a linear operator with no proper invariant subspaces

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 $$M\cap N^\perp$$.