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.