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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A decomposition of weighted translation operators

Author: Joseph J. Bastian
Journal: Trans. Amer. Math. Soc. 224 (1976), 217-230
MSC: Primary 47B37; Secondary 47A35
MathSciNet review: 0420327
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Abstract: Let (X, A, m) be a measure space and T an invertible measure-preserving transformation on X. Given $\phi$ in ${L^\infty }(X)$, define operators ${M_\phi }$ and ${U_T}$ on ${L^2}(X)$ by $({M_\phi }f)(x) = \phi (x)f(x)$ and $({U_T}f) = f(Tx)$. Operators of the form ${M_\phi }{U_T}$ are called weighted translation operators. In this paper it is shown that every weighted translation operator on a sufficiently regular measure space an be decomposed into a direct integral of weighted translation operators where almost all of the transformations in the integrand are ergodic. It is also shown that every hyponormal weighted translation operator defined by an ergodic transformation is either normal or unitarily equivalent to a bilateral weighted shift. These two results along with some results concerning direct integrals of hyponormal and subnormal operators are used to show that every hyponormal (resp. subnormal) weighted translation operator is unitarily equivalent to a direct integral of normal operators and hyponormal (resp. subnormal) bilateral weighted shifts. The paper concludes with an example.

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Keywords: Hyponormal and subnormal operator, weighted shifts, direct integrals of operators, measurable transformation
Article copyright: © Copyright 1976 American Mathematical Society