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.