  This paper establishes a theory of nonlinear spectral decompositions by
considering the eigenvalue problem related to an absolutely one-homogeneous
functional in an infinite-dimensional Hilbert space. This approach is both
motivated by works for the total variation, where interesting results on the
eigenvalue problem and the relation to the total variation flow have been
proven previously, and by recent results on finite-dimensional polyhedral
semi-norms, where gradient flows can yield spectral decompositions into
eigenvectors.
  We provide a geometric characterization of eigenvectors via a dual unit ball
and prove them to be subgradients of minimal norm. This establishes the
connection to gradient flows, whose time evolution is a decomposition of the
initial condition into subgradients of minimal norm. If these are eigenvectors,
this implies an interesting orthogonality relation and the equivalence of the
gradient flow to a variational regularization method and an inverse scale space
flow. Indeed we verify that all scenarios where these equivalences were known
before by other arguments - such as one-dimensional total variation,
multidimensional generalizations to vector fields, or certain polyhedral
semi-norms - yield spectral decompositions, and we provide further examples. We
also investigate extinction times and extinction profiles, which we
characterize as eigenvectors in a very general setting, generalizing several
results from literature.
