On preservers of pseudo spectrum of skew Jordan matrix products

Abstract

Let \(\mathcal {M}_n\) be the space of \(n \times n\) complex matrices, and for \(\varepsilon > 0\) and \(A \in \mathcal {M}_n\), let \(\sigma _\varepsilon (A)\) denote the \(\varepsilon \)-pseudo spectrum of A. Maps \(\Phi \) on \(\mathcal {M}_n\) which preserve the skew Jordan semi-triple product of matrices in a sense that

$$\sigma _\varepsilon(\Phi(A)\Phi(B)*\Phi(A))= \sigma _\varepsilon (AB*A)\quad \quad (A,B \in \mathcal {M}_n)$$

are characterized, with no surjectivity assumption on them. Analogous description is obtained for the skew Jordan product on matrices, and its variant of infinite dimension is also noted.