Necessary conditions for Schatten class localization operators

We study time-frequency localization operators of the form $A_a^{\varphi_1,\varphi_2}$, where $a$ is the symbol of the operator and $\varphi_1 , \varphi_2$ are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for $A_a^{\varphi_1,\varphi_2}\in S_p(L^2(\mathbb{R} ^d))$, the Schatten class of order $p$, is that $a$ belongs to the modulation space $M^{p,\infty}(\mathbb{R} ^{2d})$ and the window functions to the modulation space $M^1$. Here we prove a partial converse: if $A_a^{\varphi_1,\varphi_2}\in S_p(L^2(\mathbb{R} ^d))$ for every pair of window functions $\varphi_1,\varphi_2\in \mathcal{S}(\mathbb{R} ^{2d})$ with a uniform norm estimate, then the corresponding symbol $a$ must belong to the modulation space $M^{p,\infty}(\mathbb{R} ^{2d})$. 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 $p=\infty$ and $p=2$, we recapture earlier results, which were obtained by different methods.