Necessary Conditions for Schatten Class Localization Operators

We study time-frequency localization operators of the form A12 a , where a is the symbol of the operator and φ1, φ2 are the analysis and synthesis windows, respectively. It is shown in [3] that a sufficient condition for A12 a ∈ Sp(L(R)), the Schatten class of order p, is that a belongs to the modulation space Mp,∞(R2d) and the window functions to the modulation space M1. Here we prove a partial converse: if A12 a ∈ Sp(L(R)) for every pair of window functions φ1, φ2 ∈ S(R2d) with a uniform norm estimate, then the corresponding symbol a must belong to the modulation spaceMp,∞(R2d). In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p =∞ and p = 2, we recapture the results in [3], which were obtained by different methods.