Linear maps preserving the Lorentz spectrum: the $2 \times 2$ case

In this paper, a complete description of the linear maps $\phi:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$, and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $\phi(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $n\geq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in $\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $\mathbb{R}^{n}$ with $n\geq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.