ACTA issues

Nonlinear maps preserving the pseudo spectral radius of skew semi-triple products of operators

M. Bendaoud, A. Benyouness, M. Sarih

Acta Sci. Math. (Szeged) 84:1-2(2018), 39-47

Abstract. Let $\mathcal H$ be a complex Hilbert space of dimension greater than $2$, and denote by $\mathcal L(\mathcal H)$ the algebra of all bounded linear operators on $\mathcal H$. For $\varepsilon >0$ and $T\in \mathcal L(\mathcal H)$, let $r_\varepsilon (T)$ denote the $\varepsilon $-pseudo spectral radius of $T$. Let $\mathfrak {S}_1$ and $\mathfrak {S}_2$ be subsets of $\mathcal L(\mathcal H)$ which contain all rank one operators and the identity. A characterization is obtained for surjective maps $\phi \colon \mathfrak {S}_1\rightarrow \mathfrak {S}_2$ satisfying $r_{\varepsilon }(\phi (T)\phi (S)^*\phi (T))=r_{\varepsilon }(TS^*T)$ ($T, S\in \mathfrak {S}_1$). An analogous description is also obtained for the pseudo spectrum of operators.

DOI: 10.14232/actasm-017-825-8

AMS Subject Classification (1991): 47B49, 47A10, 47A25

Keyword(s): pseudo spectral radius, skew product of operators, nonlinear preservers

Received November 22, 2017, and in revised form January 12, 2018. (Registered under 75/2017.)