Interpolation d'opérateurs entre espaces de fonctions holomorphes

Let ${\displaystyle K}$ be a compact subset of an hyperconvex open set ${\displaystyle D\subset {ℂ}^n}$, forming with D a Runge pair and such that the extremal p.s.h. function ω(·,K,D) is continuous. Let H(D) and H(K) be the spaces of holomorphic functions respectively on D and K equipped with their usual topologies. The main result of this paper contains as a particular case the following statement: if T is a continuous linear map of H(K) into H(K) whose restriction to H(D) is continuous into H(D), then the restriction of T to ${\displaystyle H\left(D_{\alpha }\right)}$ is a continuous linear map of ${\displaystyle H\left(D_{\alpha }\right)}$ into ${\displaystyle H\left(D_{\alpha }\right)}$, ∀α ∈ ]0,1[ where ${\displaystyle D_{\alpha }=\{z\in D:\omega (z,D,K)<\alpha \}}$.