Abstract
We study conjugations on L^2(mathbb {T}^N) and their behaviour with respect to multiplication operators. Full characterizations of conjugations commuting or intertwining with multiplication operators are obtained. We also characterize conjugations leaving invariant subspaces being invariant for multiplication by independent variables. The subspaces not being the multiplication of the Hardy space H^2(mathbb {D}^N) by inner function are also considered.
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