  We consider K\"ahler toric manifolds $N$ that are torifications of
statistical manifolds $\mathcal{E}$ in the sense of [M. Molitor, "K\"ahler
toric manifolds from dually flat spaces", arXiv:2109.04839], and prove a
geometric analogue of the spectral decomposition theorem in which Hermitian
matrices are replaced by K\"ahler functions on $N$. The notion of "spectrum" of
a K\"ahler function is defined, and examples are presented. This paper is
motivated by the geometrization program of Quantum Mechanics that we pursued in
previous works (see, e.g., [M. Molitor, "Exponential families, K\"ahler
geometry and quantum mechanics", J. Geom. Phys. 70, 54-80 (2013)]).
