The Introduction of the book reviews the confluence of system-theory ideas and reproducing-kernel techniques which lead to representations of backward-shift invariant subspaces in the Hardy space as ranges of observability operators, and representations for forward-shift-invariant subspaces via a Beurling–Lax representer equal to the transfer function of the linear system. This pair of backward-shift-invariant and forward-shift-invariant subspaces form an in general generalized orthogonal decomposition (in the sense of de Branges–Rovnyak) of the ambient Hardy space. All this leads to the de Branges–Rovnyak model theory and characteristic operator function for a Hilbert-space contraction operator.