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.