Linear maps preserving the Lorentz-cone spectrum in certain subspaces of $M_{n}$

Abstract

In this paper, we completely characterize the linear maps $\phi :\mathcal {M} \rightarrow \mathcal {M}$ that preserve the Lorentz-cone spectrum, when $\mathcal {M}$ is one of the following subspaces of the space $M_{n}$ of $n\times n$ real matrices: the subspace of diagonal matrices, the subspace of block-diagonal matrices $\widetilde{A}\oplus [a]$, where $\widetilde{A}\in M_{n-1}$ is symmetric, and the subspace of block-diagonal matrices $\widetilde{A}\oplus [a]$, where $\widetilde{A}\in M_{n-1}$ is a generic matrix. In particular, we show that $\phi$ should be what we call a standard map, namely, a map of the form $\phi (A)=PAQ$ for all $A\in \mathcal {M}$ or $\phi (A)=PA^{T}Q$ for all $A\in \mathcal {M},$ for some matrices $P,Q\in M_{n}$. We then characterize the standard maps preserving the Lorentz-cone spectrum, when $\mathcal {M}$ is the subspace $S_{n}$ of symmetric matrices. The case $\mathcal {M=}M_{n}$ was considered in a recent paper by Seeger (LAA 2020). We include it here for completeness.