Preservers of pseudo spectra of operator Jordan triple products (Q2807236)

Let \(\mathcal{H}\) be a complex Hilbert space and \(\mathcal{L(H)}\) the algebra of all bounded linear operators on \(\mathcal{H}\). Suppose that \(\varepsilon >0\) and \(T\in \mathcal{L(H)}\). We write \(\sigma _{\varepsilon}(T)\) for the \(\varepsilon \)-pseudo spectrum of \(T,\) and \(r_{\varepsilon}(T) \) for the \(\varepsilon \)-pseudo spectral radius of \(T\).NEWLINENEWLINEThe paper under review has three sections. In the second section, the authors study (nonlinear) maps on \(\mathcal{L(H)}\) that preserve the \(\varepsilon \)-pseudo spectral radius of Jordan triple products of operators. One of the two main results of this section is as follows.NEWLINENEWLINELet \(\mathcal{H}\) be an infinite-dimensional complex Hilbert space and \(\varepsilon >0\). A surjective map \(\phi :\mathcal{L(H)\rightarrow L(H)}\) satisfies NEWLINE\[NEWLINEr_{\varepsilon }(\phi (T)\phi (S)\phi (T))=r_{\varepsilon }(TST)NEWLINE\]NEWLINE for every \(T,S\in \mathcal{L(H)}\) if and only if there is a functional \(\alpha :\mathcal{L(H)\rightarrow }\mathbb{T}\) and a bounded linear or bounded conjugate linear unitary operator \(U:\mathcal{H\rightarrow H}\) such that either \(\phi (T)=\alpha (T)UTU^{\ast }\) or \(\phi (T)=\alpha (T)UT^{\ast}U^{\ast }\) for all \(T\in \mathcal{L(H)}\). Here, \(\mathbb{T}\) denotes the unit circle of the complex field \(\mathbb{C}\).NEWLINENEWLINEThe authors give an analogous result for the finite-dimensional case without the surjectivity assumption, i.e., they characterize maps on \(\mathcal{M}_{n}\), the algebra of all \(n\times n\) complex matrices, where \(n\geq 3\), that preserve the \(\varepsilon\)-pseudo spectral radius of Jordan triple products of matrices.NEWLINENEWLINEIn the last section, the authors derive two corollaries of the above results. Namely, they study (nonlinear) preservers of the \(\varepsilon\)-pseudo spectrum of Jordan triple products of operators. For an infinite-dimensional complex Hilbert space \(\mathcal{H}\), the authors show that every surjective map \(\phi :\mathcal{L(H)\rightarrow L(H)}\) which satisfies NEWLINE\[NEWLINE\sigma _{\varepsilon }(\phi (T)\phi (S)\phi (T))=\sigma _{\varepsilon }(TST)NEWLINE\]NEWLINE for every \(T,S\in \mathcal{L(H)}\) is of a nice form. The authors again give an analogous result for the finite-dimensional case without the surjectivity assumption.NEWLINENEWLINEThe authors conclude the paper with the open question whether in some of the (above) results the Hilbert space \(\mathcal{H}\) can be replaced by a general Banach space.