Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector (Q616138)

Let \(X\) be a complex Banach space and denote by \(B(X)\) the Banach algebra of all bounded linear operators on \(X\). The local spectral radius of \(T\in B(X)\) at a vector \(x\in X\) is defined by \( r_T(x)=\limsup_{n\to \infty}\| T^n x\|^{1/n}\). It is proved that a linear and surjective map \(\varphi:B(X)\to B(X)\) which satisfies \(r_{\varphi(T)}(e)\leq r_T(e)\) for all \(T\in B(X)\) and for some fixed vector \(e\neq 0\) is automatically continuous. This gives an affirmative answer to a question asked by \textit{J.\,Bračič} and \textit{V.\,Müller} [Stud.\ Math.\ 194, No.\,2, 155--162 (2009; Zbl 1182.47004)].