On universal realizability of spectra

A list $\Lambda =\{\lambda _{1},\lambda_{2},\ldots ,\lambda _{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list $\Lambda $ is said to be universally realizable ($\mathcal{UR}$) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by $\Lambda $. It is well known that an $n\times n$ nonnegative matrix $A$ is co-spectral to a nonnegative matrix $B$ with constant row sums. In this paper, we extend the co-spectrality between $A$ and $B$ to a similarity between $A$ and $B$, when the Perron eigenvalue is simple. We also show that if $\epsilon \geq 0$ and $\Lambda =\{\lambda _{1},\lambda_{2},\ldots ,\lambda _{n}\}$ is $\mathcal{UR},$ then $\{\lambda _{1}+\epsilon ,\lambda _{2},\ldots,\lambda _{n}\}$ is also $\mathcal{UR}$. We give counter-examples for the cases: $\Lambda =\{\lambda_{1},\lambda_{2},\ldots ,\lambda _{n}\}$ is $\mathcal{UR}$ implies $\{\lambda _{1}+\epsilon ,\lambda _{2}-\epsilon ,\lambda_{3},\ldots ,\lambda_{n}\}$ is $\mathcal{UR},$ and $\Lambda _{1},\Lambda _{2}$ are $\mathcal{UR}$ implies $\Lambda _{1}\cup \Lambda _{2}$ is $\mathcal{UR}$.