Hilbert transform LTI

Question

I'm studying the Hilbert transform and its properties. I found out that Hilbert transform can be considered as a linear time-invariant system. How can I show this? I guess this should be trivial as my book does not show it.

Answer 1

A linear operator $\Theta$ is an operator acting on functions $F$ and $G$, with coefficients $a$ and $b$, such that the following equality holds:

$$ \Theta(aF+bG) = a \Theta(F) + b \Theta(G) $$

Put into words, order of operations does not matter with respect to the linear operator, multiplication of functions by coefficients, and addition of functions with each other.

To show that the Hilbert Transform is a linear operator, apply the principle given above with the Hilbert Transform in place of $\Theta$.