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Transactions of the American Mathematical Society

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Wandering vectors for irrational rotation unitary systems

Author: Deguang Han
Journal: Trans. Amer. Math. Soc. 350 (1998), 309-320
MSC (1991): Primary 46N99, 47N40, 47N99
MathSciNet review: 1451604
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Abstract: An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for wandering vector multipliers, and strengthen this by showing that for an irrational rotation unitary system $\mathcal {U}$, every unitary operator in $w^{*}(\mathcal {U})$ is a wandering vector multiplier. Moreover, we show that there is a class of wandering vector multipliers, induced in a natural way by pairs of characters of the integer group $\mathbb {Z}$, which fail to factor even as the product of a unitary in $\mathcal {U}’$ and a unitary in $w^{*}(\mathcal {U})$. Incomplete maximal wandering subspaces are also considered, and some questions are raised.

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Additional Information

Deguang Han
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Address at time of publication: Department of Mathematics, Qufu Normal University, Shandong, 273165 P.R. China

Keywords: Irrational rotation unitary system, wandering vector and subspace
Received by editor(s): March 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society