= Generalized eigenfunction in the forward-operator metric
{title2=$\lambda K^*Ku\in\partial J(u)$}
A nonzero $u$ satisfying this relation, with $Ku\ne0$, generalizes the <eigenfunction of an absolutely one-homogeneous functional> to the quadratic geometry induced by the forward map. Pairing with $u$ yields $\lambda=J(u)/\|Ku\|^2$. With unit data norm it is a <generalized singular vector>. If $\lambda>0$, $0\leq\alpha\lambda\leq1$ and the exact data are $Ku$, the vector $(1-\alpha\lambda)u$ minimizes the quadratic <variational regularization> objective. Substitute its raywise <subgradient> into the <subgradient optimality condition>. Every minimizing branch has the same predicted data, and injectivity of $K$ gives equality of vectors. The <Bregman distance> along this ray is zero even though the reconstruction is shrunk. For nonnegative $J$ the branch continues as $(1-\alpha\lambda)_+u$; nonnegativity supplies $0\in\partial J(0)$ for the threshold argument.
Background on these generalized singular vectors: https://arxiv.org/abs/1211.2057[Benning and Burger, Ground States and Singular Vectors of Convex Variational Regularization Methods].
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