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I have an operator on the half line $\mathbb{R}^+$, I know its behavior on exponential functions. In particular, given a function $g$ \begin{align*} \mathcal{A}_g e^{-tk}=g(k) e^{-tk}, \quad t>0. \end{align*} For example, if $g(x)=x^2$, we have $\mathcal{A}_{t^2} e^{-tk}=\partial_t^2 e^{-tk}=k^2 e^{-tk}$ or if $g(x)=x$ we have $\mathcal{A}_{t} e^{-tk}=-\partial_t e^{-tk}=k e^{-tk}$. We can also consider $\mathcal{A}_g$ for translation, so it is not defined only as derivatives.

I would like to extend $\mathcal{A}_g f(t)$, where $f(t)$ is a general function... I thought I would use the Laplace transforms of the functions, but the operator it is not like a convolution in my opinion (for example for derivatives I have to consider the function in $0$ when I use the Laplace transform...)

Can you help me? Thank you very much!

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