Weighted Hardy Spaces Over the Unit Ball: The Freely Noncommutative and Commutative Settings

It is known that backward-shift-invariant subspaces of the Hardy space \(H^2\) serve as the model spaces for a large class of contraction operators while forward-shift-invariant spaces (i.e., the orthogonal complements of backward-shift-invariant subspaces) have a related Beurling representation in terms of inner functions. Furthermore any such orthogonal decomposition of the whole space \(H^2\) also has a discrete-time linear-system interpretation. Recently there has been a surge of research activity elaborating on these and related results in multivariable settings, both commutative and freely noncommutative. Here we review these results with special emphasis on how additional perspective is had by looking at the commutative and freely noncommutative cases together.