= Solution
A <singular value system> of a <compact operator> consists of positive numbers $\sigma_j$ and orthonormal families $v_j\in X$, $u_j\in Y$ satisfying
$$
\boxed{Av_j=\sigma_j u_j,\qquad A^*u_j=\sigma_jv_j.}
$$
Thus $v_j$ are positive-eigenvalue <eigenvectors> of $A^*A$ and $u_j$ of $AA^*$, with <eigenvalue> $\sigma_j^2$. The families are complete in $(\ker A)^\perp$ and $\overline{\operatorname{ran}A}$ respectively. The <singular values> can be listed nonincreasingly with multiplicities, and tend to zero in the infinite-rank case. For finite rank there are only finitely many positive <singular values>; zero-kernel directions are handled separately.
Use the convention that $\langle f,g\rangle$ is conjugate-linear in its first entry. Then the <singular value system> gives
$$
Ax=\sum_j\sigma_j u_j\langle v_j,x\rangle,\qquad A^\dagger y=\sum_j\frac{\langle u_j,y\rangle}{\sigma_j}v_j.
$$
The latter series converges precisely on the admissible range component specified by the <Picard criterion>:
$$
\sum_j\frac{|\langle u_j,y\rangle|^2}{\sigma_j^2}<\infty,
$$
with $R^\perp$ components annihilated by the inverse. Merely writing a formal singular expansion does not imply it converges in $X$.
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