Say we let the operator

It is then possible to wire this operator in the form like the polar decomposition of the bounded operators on some Hilbert space.

Let be the algebra of continuous functions on the circle and consider a function with Fourier transform

Then we have

I can show that

Then defining

we can find which we can just think of as a matrix with along the diagonal

I would like to find the inverse Fourier transform of , it should resemble something that looks like a derivative, but alas I cannot find it.

Any help will be greatly appreciated