Chapter 3 introduces the notion of a contractive multiplier between weighted Hardy–Fock spaces (the analog of a Schur-class function for the classical setting). Unlike the classical case, in this general setting the notion of inner partitions into a number of distinct cases: (i) strictly inner (isometric multiplier) (ii) McCT (McCullough-Trent) inner (partially isometric multiplier), (iii) Bergman inner (contractive multiplier which is isometric when restricted to constants). For appropriately restricted pairs of input/output vectorial weighted Hardy–Fock spaces, analogs of the classical connections with dissipative/conservative linear input/state/output multidimensional linear systems, kernel decompositions, as well as corresponding generalized orthogonal decompositions of the ambient weighted Hardy–Fock space as a sum of a backward and a forward-shift-invariant subspace, are explored. These results are fundamental for the work of the succeeding Chapters.

Chapter 5 obtains Beurling–Lax-type representation theorems for isometrically included forward-shift-invariant subspaces as the range of a partially isometric (i.e., McCT-inner) multiplier coming off a (non-weighted) Hardy–Fock space. More generally, a contractively included forward-shift-invariant subspace of a weighted Hardy–Fock space is characterized as the range of a contractive multiplier coming off a (non-weighted) Hardy–Fock space with intrinsic norm equal to the lifted norm induced by the representation as the range of the associated contractive multiplier. When some additional conditions are satisfied, it is possible to obtain more explicit transfer-function realizations for the Beurling–Lax representer. These additional conditions are intimately connected with the question as to when the Brangesian complement of a forward/backward shift-invariant subspace is backward/forward shift-invariant. An example is given to show that, unlike as in the classical setting or in the case where the shift-invariant subspace is isometrically included in the ambient weighted Hardy–Fock space, it can happen that the question has a negative answer.