On complex interpolation and spectral continuity
Volume 130 / 1998
Abstract
Let $[X_0,X_1]_t$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_0$ and $X_1$ will act boundedly on each $[X_0,X_1]_t$. Let $T_t$ denote such an operator when considered on $[X_0,X_1]_t$, and $σ(T_t)$ denote its spectrum. We are motivated by the question of whether or not the map $t → σ(T_t)$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t → (σ(T_t))^∧$ (polynomially convex hull) and $t → ∂_e(σ(T_t))$ (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: $t → (σ(T_t))^∧$ is upper semicontinuous but not necessarily continuous, and $t → ∂_e(σ(T_t))$ is lower semicontinuous but not necessarily continuous.
