"Toeplitz localization operators: spectral functions density"
Hutnik, OndrejWe consider two classes of localization operators based on the Calderón and Gabor reproducing formulas. If the generating symbols depend on the first coordinate in the phase space, the Toeplitz localization operators (TLOs) exhibit an explicit diagonalization, i.e., there exists an isometric isomorphism that transforms all TLOs to the multiplication operators by some specific functions - we call them spectral functions. Using the Wiener's deconvolution technique on the real line, we prove that the set of spectral functions is dense in the C*-algebra of bounded uniformly continuous functions on the real line under the assumption that the Fourier transform of the kernel function does not vanish on the real line. This provides an explicit and independent description of the C*-algebra generated by the set of spectral functions. As a consequence we give an explicit description of the C*-algebra generated by vertical Toeplitz operators acting on true poly-analytic Bergman spaces over the upper half-plane.