# Extend an operator from exponential functions to all functions?

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!