Linear maps that preserve parts of the spectrum on pairs of similar matrices

In this paper, we characterize linear bijective maps $\varphi$ on the space of all $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have at least one common eigenvalue for each similar matrices $A$ and $B$. Using this result, we characterize linear bijective maps having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have common elements for each matrices $A$ and $B$ having the same spectrum. As a corollary, we also characterize linear bijective maps preserving the equality of the spectrum.