On Bilateral Weighted Shifts in Noncommutative Multivariable Operator Theory

We present a generalization of bilateral weighted shift operators for the noncommutative multivariable setting. We discover a notion of periodicity for these shifts, which has an appealing diagramatic interpretation in terms of an infinite tree structure associated with the underlying Hilbert space. These shifts arise naturally through weighted versions of certain representations of the Cuntz C∗-algebrasOn. It is convenient, and equivalent, to consider the weak operator topology closed algebras generated by these operators when investigating their joint reducing subspace structure. We prove these algebras have non-trivial reducing subspaces exactly when the shifts are doubly-periodic; that is, the weights for the shift have periodic behaviour, and the corresponding representation of On has a certain spatial periodicity. This generalizes Nikolskii’s Theorem for the single variable case. In [24] and [25], we began studying versions of unilateral weighted shift operators in noncommutative multivariable operator theory. We called them weighted shifts on Fock space since they act naturally on the full Fock space Hilbert space. These shifts and the algebras they generate were first studied by Arias and Popescu [3] from the perspective of weighted Fock spaces. The basic goals of this program are to extend results from the commutative (single variable) setting and, at the same time, expose new noncommutative phenomena. In the current paper, we continue this line of investigation by presenting versions of bilateral weighted shift operators for the noncommutative multivariable setting. Our analysis is chiefly spatial in nature: we examine the joint reducing subspace structure for these shifts, and consider reducibility questions for the weak operator topology closed (nonselfadjoint) algebras they generate. In particular, we give a complete characterization 2000 Mathematics Subject Classification. 47L75, 47B37, 47L55. key words and phrases. Hilbert space, operator, bilateral weighted shift, periodicity, reducing subspaces, infinite word, noncommutative multivariable operator theory, nonselfadjoint operator algebras, Fock space. 1 partially supported by a Canadian NSERC Post-doctoral Fellowship. 1