Here is the problem 152 in A Hilbert Space Problem Book.
Every contraction whose powers tend strongly to zero is unitarily equivalent to a part of adjoint of a unilateral shift.
Given a Hilbert space $H$ and on it a contraction $A$ such that $A^n \to 0$ strongly, it is shown in the proof that $VAf=U^*Vf \hspace{0.3cm}\forall f \in H$ where $V: H \to V(H) \subset \tilde{H}$ is unitary and $U: \tilde{H} \to \tilde{H}$ is unilateral shift. So $V(H)$ is an invariant subspace of $U^*$ and therefore $ A = V^*U^*|_{V(H)}V $.
My question is: Can I drop the restriction? That is, can I say $A=V^*U^*V$? Since $U^*$ in this case must act on the range of $V$, it seems to me that it makes no difference whether we restrict $U^*$ to $V(H)$ or not. So can I say $A$ is actually unitarily equivalent to the adjoint of an unilateral shift?
My question may sound silly, but I really can't figure out why is incorrect.