Reproducing spaces and localization operators (Q1887419)

The authors firstly write the operator \(T_{0}: L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{C},e^{- {1 \over 2}| z |^{2}} {dz d{\overline{z}} \over 4 {\pi} i}),\) remarking that \(T_{0}\) is an isomorphic mapping from \(L^{2}(\mathbb{R})\) to the Bargmann space \(B_{0}=\{ F \text{ analytic} : F \in L^{2}(\mathbb{C},e^{-{1 \over 2} | z |^{2}} {dz d {\overline {z}} \over {4 {\pi} i}}) \}\). They also show that \(T_{n}L^{2}(\mathbb{R})\) is a reproducing space with the reproducing kernel instead of \(T_{0}L^{2}(\mathbb{R})\) for a natural number \(n\). They give a relation between a localization operator over a reproducing space and a localization operator associated with windowed Fourier transform.