Time–Frequency analysis of localization operators

We study a class of pseudodifferential operators known as time–frequency localization operators, Anti-Wick operators, Gabor–Toeplitz operators or wave packets. Given a symbol a and two windows ϕ1,ϕ2, we investigate the multilinear mapping from (a,ϕ1,ϕ2)∈S′(R2d)×S(Rd)×S(Rd) to the localization operator Aaϕ1,ϕ2 and we give sufficient and necessary conditions for Aaϕ1,ϕ2 to be bounded or to belong to a Schatten class. Our results are formulated in terms of time–frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.