Generalized balian-low theorem in hilbert spaces

In mathematics, all the results of the uncertainty principle types can be viewed from two perspectives. First, the multiplication operator X and differentiation operator D display a property of non-commutativity. More precisely, D X − X D = 1 2 π i I . Second, a function and its Fourier transform cannot be well localized simultaneously. The two perspectives are mutually related, because the Fourier transform operator ℱ conjugates X and D That is, X = ℱDℱ *. The classical Balian-Low theorem (BLT) [2] shows that a window function generating a Gabor Riesz basis cannot be localized in both time and frequency. So it is the result of the uncertainty principle types for Gabor systems. Inspired by [6], we generalize the Balian-Low theorem to general Hilbert spaces.