Automatic continuity for linear surjective maps compressing the local spectrum at fixed vectors

Abstract:

Let $X$ and $Y$ be complex Banach spaces and denote by $\mathcal {L}(X)$ and $\mathcal {L}(Y)$ the algebras of all bounded linear operators on $X$, respectively $Y$. Let also $x_0 \in X$ and $y_0 \in Y$ be nonzero vectors. We prove that if $\varphi : \mathcal {L}(X) \rightarrow \mathcal {L} (Y)$ is a linear surjective map such that for each $T \in \mathcal {L}(X)$ we have that the local spectrum of $\varphi (T)$ at $y_0$ is a subset of the local spectrum of $T$ at $x_0$, then $\varphi$ is automatically continuous. We also give a new proof for the automatic continuity of linear surjective maps decreasing the local spectral radius at some fixed nonzero vector. As a corollary, we obtain that the characterizations of J. Bračič and V. Müller for linear surjective maps on $\mathcal {L}(X)$ preserving the local spectrum/local spectral radius at some fixed vector can be obtained with no continuity assumptions on them.