Phase-shift a Broadband Time-Domain Function
A particular example of your operator is \$\phi=-\pi/2\$ which produces the Hilbert Transform. You could implement it by taking the Fourier transform and phase shifting each component by something which kind of depends on \$\omega\$... negative frequencies are shifted by the opposite phase. This really is a non-trivial operation because you need to integrate over \$\omega\$ to recover the original signal. The problem is that this process is not causal. I would suggest searching for &quot;zero-phase digital filter&quot; or &quot;digital Hilbert transform&quot;; you will find many techniques for achieving approximate constant phase shift in a particular pass band with either FIR or IIR approaches. For instance, JOS has some material.
More generally, one can define an all-pass filter as any filter with unity gain at all frequencies.