Solution
= Solution
For $d_j=1/j$, the equation $Du=f$ forces $u_j=jf_j$. It has a solution in $\ell^2$ exactly when
$$
\sum_{j=1}^\infty j^2|f_j|^2<\infty.
$$
For example, $f_j=1/j$ defines an element of $\ell^2$, but its forced preimage is the constant sequence $u_j=1$, which is not in $\ell^2$. Thus existence can fail. Since every $d_j$ is nonzero, $D$ has <trivial kernel> and any solution is unique.