Linear maps preserving the Lorentz-cone spectrum in certain subspaces of M<inf>n</inf>

In this paper, we completely characterize the linear maps ϕ: M→ M that preserve the Lorentz-cone spectrum, when M is one of the following subspaces of the space Mn of n× n real matrices: the subspace of diagonal matrices, the subspace of block-diagonal matrices A~ ⊕ [ a] , where A~ ∈ Mn-1 is symmetric, and the subspace of block-diagonal matrices A~ ⊕ [ a] , where A~ ∈ Mn-1 is a generic matrix. In particular, we show that ϕ should be what we call a standard map, namely, a map of the form ϕ(A) = PAQ for all A∈ M or ϕ(A) = PATQ for all A∈ M, for some matrices P, Q∈ Mn. We then characterize the standard maps preserving the Lorentz-cone spectrum, when M is the subspace Sn of symmetric matrices. The case M= Mn was considered in a recent paper by Seeger (LAA 2020). We include it here for completeness.