Vector valued eigenfunctions of ergodic transformations

Flytzanis, E.

doi:10.1090/S0002-9947-1978-0499076-8

Vector valued eigenfunctions of ergodic transformations
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Trans. Amer. Math. Soc. 243 (1978), 53-60 Request permission

Abstract:

We study the solutions X, T, of the eigenoperator equation \[ X(h( \cdot )) = TX( \cdot ) {\text {a}}{\text {.e}}{\text {.}}\], 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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