Chapter 9 is also concerned with weighted Hardy–Fock spaces but with the associated kernel function for the weighted Hardy–Fock space having a certain prescribed form. The resulting weight sequence satisfies the admissibility requirements imposed in earlier chapters only in special cases. Consequently, the results (and associated formulas) parallel the results from earlier chapters, but with the associated formulas having a more complicated form involving explicitly the parameters appearing in the prescribed kernel function. The resulting operator-model theory generalizes earlier such results appearing in the literature for the commutative case.

Chapter 6 deals with two flavors of Beurling–Lax representations which have been studied in the context of Bergman spaces: (i) a Beurling–Lax-type representation based on the shift-invariant subspace being generated by a so-called quasi-wandering subspace, and (ii) a Beurling–Lax representation based on the shift-invariant subspace being generated (but non-orthogonally) by its canonically defined wandering subspace.

Chapter 7 obtains a Beurling–Lax representation for an isometrically included forward-shift-invariant subspace of a weighted Hardy–Fock space which involves a whole family of Bergman-inner-like multipliers (rather than a single inner multiplier) which leads to an orthogonal decomposition of the forward-shift-invariant subspace. The Bergman-inner family can be viewed as the transfer-function family for a time-varying noncommutative, multidimensional weighted-conservative input/state/output linear system, and can be viewed as a time-varying isometric multiplier from a time-varying unweighted Hardy–Fock space to the given weighted Hardy–Fock space.

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.