Spectral set of a linear system with discrete time

Fix a certain class of perturbations of the coefficient matrix $A(\cdot)$ of a discrete linear homogeneous system of the form
\begin{equation*} x(m+1)=A(m)x(m),\quad m\in\mathbb N,\quad x\in\mathbb R^n, \end{equation*}
where the matrix $A(\cdot)$ is completely bounded on $\mathbb N$. The spectral set of this system corresponding to a given class of perturbations is the collection of complete spectra of the Lyapunov exponents of perturbed systems when perturbations runs over the whole class considered. In this paper, we examine the class ${\mathcal R}$ of multiplicative perturbations of the form
\begin{equation*} y(m+1)=A(m)R(m)x(m),\quad m\in\mathbb N,\quad y\in\mathbb R^n, \end{equation*}
where the matrix $R(\cdot)$ is completely bounded on $\mathbb N$. We obtain conditions that guarantee the coincidence of the spectral set $\lambda({\mathcal R})$ corresponding to the class ${\mathcal R}$ with the set of all nondecreasing tuples of $n$ numbers.