My question is as follows: For $k\in\mathbb{N}$, $k\ge 2$, I need to find an operator $A: L^2(\mathbb{R}\to L^2(\mathbb{R})$ such that for all $u\in L^2(\mathbb{R})$ we have $[(A^* A)(u)](t)=\left|t\right|^k.u(t)$ for almost $t\in\mathbb{R}$. Here $A^*$ denotes the adjoint operator of $A$. Actually, I do not sure on the existence of $A$ in my question. Can someone help me?
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2$\begingroup$ Maybe the self-adjoint operator $A(u)(t) = |t|^{k/2} u(t)$? $\endgroup$– Pedro M.Oct 6, 2014 at 16:42
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$\begingroup$ I understand your hint. Thank you very much. $\endgroup$– CaoOct 6, 2014 at 16:58
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$\begingroup$ However, $|t|^k u(t)$ is not guaranteed to be in $L^2(\mathbb{R})$; maybe you mean $|t|^{-k} u(t)$? If this is the case, my suggestion will work for $k \geq 4$, but not for $k=2$ and $k=3$. $\endgroup$– Pedro M.Oct 6, 2014 at 17:30
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$\begingroup$ There is a good chance that $A$ is only supposed to be a closed operator, in which case your suggestion works. $\endgroup$– Jonas DahlbækOct 6, 2014 at 18:00
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