We will talk about operator representation of frames, that is the possibility to write a frame of a Hilbert space as orbit of an operator on a single element. The iteration of the operator can be over the natural numbers or over all the integer numbers. The existence of such a representation is equivalent to the linear independence of the frame, but the operator might be unbounded. Necessary and sufficient conditions are known to guarantee that the operator is bounded and they are related to shift-invariance of the kernel of the synthesis operator. In particular, we will focus on an open question, regarding the possibility of representing a Gabor frame as orbit over Z of a bounded operator.