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

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

“…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 ≥ 3 obtained by Bueno, Furtado, and Sivakumar (2021). The case n = 2 has some specificities, when compared to the case n ≥ 3, due to the fact that the Lorentz cone in R 2 is polyedral, contrary to what happens when it is contained in R n with n ≥ 3.…”

“…Linear preservers of the L-spectrum. In [3] the following important result was shown for matrices of size n ≥ 3, although the presented proof is also valid for 2 × 2 matrices. By W n we denote any of the spaces M n or S n , the subspace of symmetric matrices.…”

“…For completeness and for purpose of comparison with our main result, Theorem 1.1, we next state the characterization obtained in [3] of the standard linear maps φ : W n → W n that preserve the L-spectum, when n ≥ 3.…”

“…In [3] the authors focused on the problem of studying the linear maps φ : W n → W n that preserve the L-spectrum, that is, such that σ K (φ(A)) = σ K (A), for all A ∈ W n , where W n is a subspace of M n and n ≥ 3. The authors started by characterizing such maps φ for the following subspaces W n of M n : the subspace of diagonal matrices; the subspace of block-diagonal matrices A ⊕ [a], where A ∈ M n−1 is symmetric; and the subspace of block-diagonal matrices A ⊕ [a], where A ∈ M n−1 is a generic matrix.…”

“…Also, for both cases of W 2 , we show that the nature of the L-eigenvalues (being Linear maps preserving the Lorentz spectrum: the 2 × 2 case associated with L-eigenvectors in the interior or on the boundary of the Lorentz cone) is not changed by the L-spectrum preservers. To prove our results, we introduce techniques that explore the knowledge of the Lorentz spectrum of matrices in M 2 and hope that the ideas behind our proofs can be extended to complete the study of the L-spectrum preservers studied in [3] for matrices in M n , with n ≥ 3, in which case no connection exists with the Pareto cone. It follows from our characterization that such preservers on M 2 are standard and that, in the case W 2 = M 2 , their form is less restrictive than the one for n ≥ 3.…”