$\DeclareMathOperator\Hom{Hom}$Let $G$ be a (pro-)cyclic group (topologically) generated by $\phi$, and let $V,W$ be two finite-dimensional (continuous) $k$-linear semisimple representations of $G$, where $k$ is some algebraically closed (complete) field.

Let $$P_V(t) = \prod_{i=1}^n (t - a_i), P_W(t) = \prod_{j=1}^m (t - b_j)$$ be the characteristic polynomials of $\rho_V(\phi), \rho_W(\phi)$ respectively.

I would like to know if $$\dim_k \Hom_G(V, W) = \# \{ (i, j) : a_i = b_j \}$$

I know that $\dim_k \Hom_G(V, W) = \langle \chi_V, \chi_W \rangle$, and that $f \in \Hom_G(V, W)$ iff $f \circ \rho_V(\phi) = \rho_W(\phi) \circ f$. But I don't know how to proceed from there.

(NB: already asked here two weeks ago, with no success).