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Transactions of the American Mathematical Society

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



Vector valued eigenfunctions of ergodic transformations

Author: E. Flytzanis
Journal: Trans. Amer. Math. Soc. 243 (1978), 53-60
MSC: Primary 28A65
MathSciNet review: 0499076
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Abstract: We study the solutions X, T, of the eigenoperator equation

$\displaystyle X(h( \cdot ))\,= \,TX( \cdot )\,{\text{a}}{\text{.e}}{\text{.}}$

, where h is a measurable transformation in a $ \sigma $-finite measure space $ (S,\Sigma ,m)$, T is a bounded linear operator in a separable Hilbert space H and $ X:S \to H$ is Borel measurable. We solve the equation for some classes of measure preserving transformations. For the general case we obtain necessary conditions concerning the eigenoperators, in terms of operators induced by h in the scalar function spaces over the measure space. Finally we investigate integrability properties of the eigenfunctions.

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Keywords: Eigenvalue, measure preserving transformation, eigenoperator, vector valued eigenfunction, measures in Hilbert space, weighted shift operators
Article copyright: © Copyright 1978 American Mathematical Society

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