= Solution
The <Picard criterion> with the input/output convention above states
$$
\boxed{f\in\mathcal R(K)\iff f\in\overline{\mathcal R(K)}\ \text{and}\ \sum_j\frac{|\langle f,v_j\rangle|^2}{\sigma_j^2}<\infty.}
$$
Necessity follows from $\langle f,v_j\rangle=\sigma_j\langle u,u_j\rangle$ and <Bessel inequality>. Conversely the summability constructs $u=\sum_j\sigma_j^{-1}\langle f,v_j\rangle u_j\in U$; the closure condition ensures that its image is all of $f$. The closure condition must not be omitted: a component orthogonal to the range cannot be reconstructed.
For the diagonal example, $f_j=j^{-p}$ is itself in $\ell^2$ only for $p>1/2$. Its preimage would be $u_j=j^{1-p}$, so the range criterion is the convergence of $\sum_jj^{2-2p}$. The <P-series> gives
$$
\boxed{f\in\mathcal R(K)\iff p>3/2.}
$$
At $p=3/2$ the reconstruction <norm> diverges harmonically. For $1/2<p\leq3/2$ the data are legitimate but have no preimage; for $0<p\leq1/2$ they are not even in the specified data space.
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