 Solution

Interpret the normal-operator source condition as the displayed intersection condition: there is $v\in U$ with $w=K^{\ast }Kv\in \partial J(u^{†})$. Fix any $\alpha >0$ and put $\bar{u}=u^{†}+\alpha v$. The subgradient optimality condition for the artificial-data objective isIt holds because its first term equals $-\alpha K^{\ast }Kv=-\alpha w$. By convexity, this is sufficient for global minimality, proving the forward implication. Conversely, if $u^{†}$ minimizes that objective, the same optimality condition givesTaking $v=(\bar{u}-u^{†})/\alpha$ proves the intersection condition. Thus the two conditions are equivalent when the source vector is allowed to be zero.

The printed additional requirement $v\ne 0$ is not equivalent in general. For $K=I$ and $J\equiv 0$, the displayed source condition holds, and choosing $\bar{u}=u^{†}$ makes $u^{†}$ the minimizer, but $\partial J(u^{†})=\{0\}$ forces $v=0$. A nonzero-source version needs an additional hypothesis, such as requiring $w\ne 0$; it cannot be inferred from the printed intersection alone.