= Solution
Take $F(x)=\tfrac12\|Ax-y\|_2^2$, with $\nabla F(x)=A^T(Ax-y)$ and Lipschitz constant $L=\|A\|_{2\to2}^2$, and $H(x)=\|x\|_1$. The <proximal operator> separates over coordinates. Solving the scalar <subgradient optimality condition> $0\in\partial|z|+(z-v)/\tau$ gives <soft thresholding> $S_\tau(v)=\operatorname{sgn}(v)(|v|-\tau)_+$. Therefore the <iterative soft-thresholding algorithm> is
$$
\boxed{x^{k+1}=S_\tau\bigl(x^k-\tau A^T(Ax^k-y)\bigr),\qquad0<\tau\le\|A\|_{2\to2}^{-2}.}
$$
Here <soft thresholding> is applied coordinatewise. The objective is <convex> and coercive because of its $\ell^1$ term, so minimizers exist and the <proximal gradient method> converges to a minimizer. Uniqueness need not hold if $A$ has a nontrivial <null space>. If $A=0$, the unique minimizer is zero and any positive step size is admissible.
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