We shall discuss pseudo-differential operators with band-limited symbol. These operators play an important role in science and engineering, for example, they are commonly used to model time-variant channels in communications engineering. In this scenario, the support restriction on the (symplectic) Fourier transform of the Kohn-Nirenberg symbol, i.e., on the so-called spreading function, represents restrictions on time-delay and Doppler shift.

In recent work, we realized a strong correspondence of such operators with so-called Gabor-frame operators and, using this, we proved a classical conjecture of Thomas Kailath (1962) on the identifiability of so-called underspread operators.