Chapter 2 develops the basic notion of Formal Noncommutative Reproducing-Kernel Hilbert Space which generalizes the classical notion of Reproducing Kernel Hilbert Space to a noncommutative setting. This formalism in turn is a convenient formalism for modeling freely noncommutative shift-operator tuples.

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 4 is concerned with the state/output part of a noncommutative linear system and the range of the associated observability operator. Specifically, (i) observability operators having range landing inside of a given weighted Hardy–Fock space are characterized by the existence of a solution to certain Linear Matrix Inequality (Linear Operator Inequality in general) called a Stein inequality, (ii) conversely, subspaces of a given weighted Hardy–Fock space arising as the range of a contractive observability operator are characterized as contractively included backward-shift-invariant subspaces of the ambient Hardy–Fock space having some additional natural structural properties.

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 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 8 presents a de Branges–Rovnyak-type model theory for a given operator-tuple in an appropriate class (indexed by an admissible weight ?) of hypercontractive operator tuples. Application of results from Chapter 4 leads to a shift-type model-operator-tuple acting on a backward-shift-invariant contractively included subspace of a ?-weighted Hardy–Fock space. In the nicest case one can use results of Chapter 5 to define a characteristic operator function from which one can recover in the model the original ?-hypercontractive operator tuple up to unitary equivalence. A particular instance of this nice situation is the case where the ?-hypercontractive operator tuple is pure, or equivalently, when the backward-shift-invariant subspace is isometrically included in the ambient weighted Hardy–Fock space. In this case, there is also an alternative model theory based on a characteristic Bergman-inner family which makes use of results from Chapter 7. In this case, there is also a Bergman-inner characteristic function which is a partial unitary invariant and relates to the work on the Bergman shift from the 1990s.