Spectral distances on RCD spaces

We provide relationships between the spectral convergences in Bérard–Besson–Gallot sense, in Kasue–Kumura sense and the measured Gromov–Hausdorff convergence, for compact finite dimensional \({{\,\textrm{RCD}\,}}\) $\text{RCD}$ spaces. As an independent interest, a canonical (spectral) approximation map between such spaces constructed by given spectral data is obtained.