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Operators commuting with translations, and systems of difference equations

Miklós Laczkovich (1999)

Colloquium Mathematicae

Let = f : : f i s b o u n d e d , and = f : : f i s L e b e s g u e m e a s u r a b l e . We show that there is a linear operator Φ : such that Φ(f)=f a.e. for every f , and Φ commutes with all translations. On the other hand, if Φ : is a linear operator such that Φ(f)=f for every f , then the group G Φ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every f ( x ) = e c x , then G Φ must...

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