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It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator, while forward shift-invariant subspaces have a representation in terms of an inner function. We discuss several variants of these statements in the context of weighted Bergman spaces on the unit disk.

Weighted Bergman Spaces: Shift-Invariant Subspaces and Input/State/Output Linear Systems