Properties of the operator $T: f\to f*g$

Question

Let g be the characteristic function of [-1/2,1/2].

$T: f\to f*g$ (convolution).

I have managed to prove that T is a linear,bounded,self adjoint,injective operator and it's immage is inclused in the space $H^1(R)$. Can anyone help me to find out if T is also compact/surjective? Thank you in advance.

Answer 1

It is much easier to do this calculation in Fourier space: $$ \widehat{Tf}(\xi) = \hat g(\xi) \hat g(\xi) .$$ And the Fourier transform is an isometry on $L^2$. So look at the operator $$ S f(x) = \hat g(x) f(x) $$ and derive all the properties that way instead.