The problem about K-theory is from Rørdam's book An Introduction to K-theory for $C^*$-algebras Exercise 9.6.
Let $H$ be an infinite dimensional separable Hilbert space. Consider the short exact sequence $$ 0 \longrightarrow \mathcal{K} \stackrel{\iota}{\longrightarrow} B(H) \stackrel{\pi}{\longrightarrow} \mathcal{Q}(H) \longrightarrow 0, $$ and let $\delta_{1}: K_{1}(\mathcal{Q}(H)) \rightarrow K_{0}(\mathcal{K})$ be its associated index map.
(i) Let $E, F$ be projections in $B(H)$ such that $\operatorname{dim}(E(H)) \leqslant \operatorname{dim}(F(H))$. Show that there exists a partial isometry $V$ in $B(H)$ satisfying $V^{*} V=E$ and $V V^{*} \leqslant F$.
(ii) Let $u$ be a unitary element in $\mathcal{Q}(H)$. Show that there is $S$ in $B(H)$ such that $\pi(S)=u$ and such that $S$ is either an isometry, i.e., $S^{*} S=I$, or a co-isometry, i.e., $S S^{*}=I$.
(iii) Let again $u$ be a unitary element in $\mathcal{Q}(H)$. Show that $u$ lifts to a unitary operator in $B(H)$ (i.e., there is a unitary operator $V$ on $H$ such that $\pi(V)=u)$ if and only if $\delta_{1}\left([u]_{1}\right)=0$.
(Here $\mathcal{K}$ is the $C^*$-algebra of compact operators in $B(H)$ and $\mathcal{Q}(H)) = B(H)/\mathcal{K}$ is the Calkin algebra, and $\delta_1$ is the index map connecting the $K_1$ and $K_0$ group.)
I have done the first two of them. The problem I face is the other half of (iii) where I have no idea of how to find a unitary lift $V \in B(H)$ such that $\pi(V)=u)$ given that $\delta_{1}\left([u]_{1}\right)=0$.
Any help is apprecited.
Use (b) to find an isometry/co-isometry $S$ mapping to $u$. Then $S$ is a Fredholm operator whose index equals $\delta([u]_1)$, once $K_0(\mathcal K)$ is identified with $\mathbb Z$.
Letting $E=1-S^*S$, and $F=1-SS^*$, we then have that $E$ and $F$ are projections of finite rank and $$ 0=\delta([u]_1)=\text{ind}(S)=$$$$=\text{rank}(F)-\text{rank}(E), $$ so $E$ and $F$ have the same rank.
Since either $E$ of $F$ is zero we deduce that both are zero, and hence that $S$ is unitary.