An extension of localization operators

We review at first the role of localization operators as a meeting point of three different areas of research, namely: signal analysis, quantization and pseudo-differential operators. We extend then the correspondence between symbol and operator which characterizes localization operators to a more general situation, introducing the class of bilocalization operators. We show that this enlargement yields a quantization rule that is closed under composition. Some boundedness results are deduced both for localization and bilocalization operators. In particular for bilocalization operators we prove that square integrable symbols yield bounded operators on \(L^2\) $L^2$ and that the class of bilocalization operators with integrable symbols is a subalgebra of bounded operators on every fixed modulation space.