Linear maps preserving the Lorentz-cone spectrum in certain subspaces of $$M_{n}$$
Linear maps preserving the Lorentz-cone spectrum in certain subspaces of $$M_{n}$$
复制标题
线性映射在 $$M_{n}$$ 的某些子空间中保留洛伦兹锥频谱
DOI：
--
复制
发表时间：
2021
影响因子：
1.2
通讯作者：
K. Sivakumar
中科院分区：
文献类型：
--
作者：
M. I. Bueno;S. Furtado;K. Sivakumar
In this paper, we completely characterize the linear maps $$phi :mathcal {M}
ightarrow mathcal {M}$$
that preserve the Lorentz-cone spectrum, when $$mathcal {M}$$
is one of the following subspaces of the space $$M_{n}$$
of $$n imes n$$
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 $$Ain mathcal {M}$$
or $$phi (A)=PA^{T}Q$$
for all $$Ain mathcal {M},$$
for some matrices $$P,Qin M_{n}$$
. 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.
