Suppose that for $A$ $\in \mathbb{R}^{m\times n}$ the singular value decomposition of $A$ is $UBV^T$ where $U$ $\in \mathbb{R}^{m\times t}$ and $B$ $\in \mathbb{R}^{t\times t}$ and $V$ $\in \mathbb{R}^{n\times t}$. If $W$ is a matrix in $\in \mathbb{R}^{m\times n}$ with $||W||_{op}$ $\leq 1$, $U^{T}W =0$ and $WV =0$, then, we are required to show that $||UV^{T}+W||_{op} \leq 1$.
I consider $U_1B_1V_1$, the singular value decomposition of $UV^{T}+W$ and I found two relations:
$UU^T +WW^T = U_1B_1^2U_1^T$ and $VV^T +W^TW = V_1B_1^2V_1^T$. But I think these relations just gives me $||UV^{T}+W||_{op} \leq \sqrt{2}$.