Peripheral local spectrum preservers and maps increasing the local spectral radius (Q2807244)

Let \(X\) and \(Y\) denote infinite-dimensional complex Banach spaces and let \(\mathcal{B}(X ,Y)\) be the space of all bounded linear operators from \(X\) to \(Y\). When \(X =Y\), we write \(\mathcal{B}(X)\) instead of \(\mathcal{B}(X ,Y)\). The spectral radius of \(T \in \mathcal{B}(X)\) will be denoted by \(r(T)\). Let \(\sigma _{T}(x)\) and \(r_{T}(x)\) denote the local spectrum and the local spectral radius of \(T \in \mathcal{B}(X)\) at \(x \in X\), respectively. The set NEWLINE\[NEWLINE\gamma _{T}(x)=\left \{\lambda \in \sigma _{T}(x): | \lambda | =r_{T}(x)\right \}NEWLINE\]NEWLINE is called the peripheral local spectrum of \(T\).NEWLINENEWLINEThe paper has six sections. In Section 2, the authors state their main results and in Section 3, they prove some auxiliary results. The main results are proved in the last three sections. The authors first describe the structure of all surjective maps on \(\mathcal{B}(X)\) that preserve the peripheral local spectrum at a nonzero fixed vector of product of operators. Next, the authors characterize maps preserving the peripheral local spectrum at a fixed vector of triple product of operators. The main tools in the proofs of these results are the characterization of the linear independence of two operators, and the characterization of rank one operators in terms of the peripheral local spectrum at a nonzero fixed vector of product or triple product of operators. Lastly, the authors turn their attention to linear maps increasing the local spectral radius at a nonzero fixed vector of \(X\) and prove the following result:NEWLINENEWLINETheorem. Let \(x_{0} \in X\) be a fixed nonzero element and let \(\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)\) be a surjective linear map. If there exists a constant \(M >0\) such that NEWLINE\[NEWLINEr_{T}(x_{0}) \leq Mr(\varphi (T))NEWLINE\]NEWLINE for all \(T \in \mathcal{B}(X)\), then \(\varphi \) is a continuous bijective map spectrally bounded from below.