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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Email: D.Han@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02065-0
PII: S 0002-9947(98)02065-0
Keywords: Irrational rotation unitary system, wandering vector and subspace
Received by editor(s): March 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society