Question about Hilbert transform
This question is hard to answer because it depends on your definition of "transform". The Hilbert transform can be written as a convolution with the kernel

$$h(t)=\frac{1}{\pi t}\tag{1}$$

and, consequently, it can be represented by a linear time-invariant (LTI) system with an impulse response given by $(1)$. So the application of any LTI system to a signal can be seen as a transform or mapping which doesn't change the domain of the signal. E.g., the application of an ideal band pass filter to an input signal maps that signal to the space of band-limited functions, and in this sense, it is a transform(ation).