Condition spectra of special operators and condition spectra preservers (Q504868)

Let \({\mathcal B}(H)\) be the algebra of bounded operators on a complex Hilbert space \(H\) and let \(\sigma(T)\subseteq{\mathbb C}\) be the spectrum of \(T\in {\mathcal B}(H)\). With each fixed real number \(\varepsilon\in(0,1)\), one defines the \(\varepsilon\)-condition spectrum and the \(\varepsilon\)-condition spectral radius of \(T\in{\mathcal B}(H)\) by \[ \sigma_{\varepsilon}(T)=\sigma(T)\cup\{\lambda\in{\mathbb C}\setminus\sigma(T);\ \|\lambda-T\|\cdot\|(\lambda-T)^{-1}\|\geq \varepsilon^{-1}\}\text{ and } r_{\varepsilon}(T)=\sup_{\lambda\in\sigma_{\varepsilon}(T)}|\lambda|. \] The authors derive a formula the for \(\varepsilon\)-condition spectrum and the \(\varepsilon\)-condition spectral radius of selfadjoint operators and of rank-one nilpotent operators. For example, \(\sigma_{\varepsilon}(P)\) of a selfadjoint projection \(P\) is a union of two non-overlapping disks centred at the real axis and \(r_{\varepsilon}(P)=\frac{1}{1-\varepsilon}\). It is shown in the sequel that a surjective map \(\phi\) on \({\mathcal B}(H)\) which preserves the \(\varepsilon\)-condition spectral radius of the product of operators in the sense that \[ r_\varepsilon(\phi(S)\phi(T))=r_\varepsilon(ST) \] is an (anti)unitary similarity modulo a unimodular valued function. If \(H\) is finite-dimensional with \(\dim H\geq 3\), one can even drop the surjectivity assumption. A similar result is derived for the Jordan triple product \(TST\) instead of an ordinary product \(ST\); then the adjoint operation is also possible. As an application, the authors classify preservers of the \(\varepsilon\)-condition spectrum of a product or a Jordan triple product (i.e., \(r_\varepsilon\) replaced by \(\sigma_\varepsilon\)). The conclusions are similar, except that a unimodular valued function takes a constant unimodular value.