Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space. Suppose that $\mathcal{H}_0\subset\mathcal{H}$ which is a dense proper subspace is the range of some bounded linear operator $T$. It is known that there are many bounded linear operator $X$ preserving $\mathcal{H}_0$ invariant.

Now given disjoint dense proper subspace $\mathcal{H}_1$ and $\mathcal{H}_2$ of $\mathcal{H}$. If there are two bounded linear operators $T_1$ and $T_2$ such that $\mathcal{H}_{i}=Ran(T_i)$ and $X \mathcal{H}_i\subset\mathcal{H}_i$, does the bounded operator $X$ need to be the form $\lambda I$ where $\lambda$ is a complex number?