Nonlinear maps preserving the inner local spectral radius

Abstract

Let \(\fancyscript{B}(X)\) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space \(X\), and let \(\iota _T(x)\) denote the inner local spectral radius of an operator \(T\in \fancyscript{B}(X)\) at any vector \(x\in X\). We characterize surjective maps on \(\fancyscript{B}(X)\) satisfying $$\begin{aligned} \iota _{T\pm S}(x)=0 \quad \text { if and only if } \quad \iota _{\varphi (T)\pm \varphi (S)}(x)=0, \end{aligned}$$ for all \(x\in X\) and \(S,~T\in \fancyscript{B}(X)\). We also determine the form of all bicontinuous bijective maps on \(\fancyscript{B}(X)\) preserving the inner local spectral radius of the difference and sum operators at a nonzero fixed vector.