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Suppose I have a linear operator $L$ s.t. $Ly(x-a)=\exp(-iax)Ly(x)$ and $L^2y(x)=ky(-x)$ where $a, k\in \mathbb R.$ Then what is $L^2y(x-a)$? May be it is really straightforward, but I don't know how to combine them!  Thank you.

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    @DavideGi$r$audo: Yes, it is the space of functions.2011-11-16

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$(L^2y)(x-a)=z(x-a)$ with $z=L^2y$ hence $z(x)=(L^2y)(x)=ky(-x)$ for every $x$ and $z(x-a)=ky(-(x-a))$, that is $(L^2y)(x-a)=ky(a-x)$.