Let U$_i$ $\in$ $\Bbb R^{n×v_i}$ be a matrix composed of unit orthogonal vectors, U = diag{U$_i$} i $\in$ {1,2,...N}. For instance \begin{matrix} U_1=\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \qquad U_2 =\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & 0 \end{bmatrix} \end{matrix} then $$ U =\left [ \begin{matrix} 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ 0 & 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & 0 & 0 & 0 & 0 \end{matrix} \right ] $$
So is that true for any matrix $\phi$ whose spectrum radius is less than 1, the spectrum radius of $U^ \mathrm{T}$ $\phi$ U is less than one. i.e. $\rho$($\phi$) < 1 $\Longrightarrow$ $\rho$($U^ \mathrm{T}$ $\phi$ U) < 1
Here is a counter example: $$ U=\pmatrix{1 \cr 0},\quad\text{and}\quad \phi=\pmatrix{2 &2 \cr -2 & -2}. $$ We have that $\rho(\phi)=0$, but $\rho(U^T\phi U)=2$.