A note on the spectral mapping theorem for point spectrum

The spectral mapping theorem, in conjunction with the concept of a spectral set, is utilized to construct reducing pairs of subspaces for a given operator. By leveraging the theory of functional calculus, we revisit the well-established spectral mapping theorem for various spectra, with a particular focus on the point spectrum. Concluding this discussion, we present an application wherein we collect and establish properties of the point spectrum of a compact exponentially m-isometry. Also, we examine the behavior of solutions of a system of differential equations associated with exponentially m-isometric matrices.