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

DOI:
https://doi.org/10.1090/S0002-9947-98-02065-0

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

Keywords:
Irrational rotation unitary system,
wandering vector and subspace

Received by editor(s):
March 11, 1996

Article copyright:
© Copyright 1998
American Mathematical Society