Christoph Neuner: Shift-invariant linear relations on the Hardy space H²

We consider the Hardy space H² on the disk. Then it is possible to show that a bounded linear operator commutes with the shift operator if and only if it is a multiplication operator. Taking this as a basis, we examine the generalisation of this theorem to higher dimensions, i.e. involving Hardy spaces of vector-valued functions and matrix-valued multiplication operators. This then allows us to characterise those subspaces of (H²)², i.e. linear relations, that are invariant with respect to the action of a shift operator.