On complex interpolation and spectral continuity

Let ${\displaystyle {[X_0,X_1]}_t}$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both ${\displaystyle X_0}$ and ${\displaystyle X_1}$ will act boundedly on each ${\displaystyle {[X_0,X_1]}_t}$. Let ${\displaystyle T_t}$ denote such an operator when considered on ${\displaystyle {[X_0,X_1]}_t}$, and ${\displaystyle \sigma \left(T_t\right)}$ denote its spectrum. We are motivated by the question of whether or not the map ${\displaystyle t\to \sigma \left(T_t\right)}$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: ${\displaystyle t\to {(\sigma \left(T_t\right))}^{\wedge }}$ (polynomially convex hull) and ${\displaystyle t\to {\partial }_e(\sigma \left(T_t\right))}$ (boundary of the polynomially convex hull). We show that the first of these maps is always upper semicontinuous, and the second is always lower semicontinuous. Using an example from [5], we now have definitive information: ${\displaystyle t\to {(\sigma \left(T_t\right))}^{\wedge }}$ is upper semicontinuous but not necessarily continuous, and ${\displaystyle t\to {\partial }_e(\sigma \left(T_t\right))}$ is lower semicontinuous but not necessarily continuous.