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We present a generalization of bilateral weighted shift operators for thenoncommutative multivariable setting. We discover a notion of periodicity forthese shifts, which has an appealing diagramatic interpretation in terms of aninfinite tree structure associated with the underlying Hilbert space. Theseshifts arise naturally through weighted versions of certain representations ofthe Cuntz C$^*$-algebras $O_n$. It is convenient, and equivalent, to considerthe weak operator topology closed algebras generated by these operators wheninvestigating their joint reducing subspace structure. We prove these algebrashave non-trivial reducing subspaces exactly when the shifts aredoubly-periodic; that is, the weights for the shift have periodic behaviour,and the corresponding representation of $O_n$ has a certain spatialperiodicity. This generalizes Nikolskii's Theorem for the single variable case.

On bilateral weighted shifts in noncommutative multivariable operator theory