- We consider identification of operator families defined via a time-frequency series expansion of the operator spreading function. The identification problem is transformed into an infinite-dimensional linear algebra problem. Our aim is to establish a connection between the identifiability of the operator family and a density measure of the time-frequency index set. In this way, the identification problem can be compared to the classical density condition for existence of Gabor frames. The conclusion is that the relationship between identifiability of such operator families and the critical density is highly intricate because of the presence of additional conditions. Criteria for identifiability are developed for families of time-frequency localized operators defined via time-frequency series expansions of the spreading function based on the Gaussian function.