Linear isometries of finite codimensions on Banach algebras of holomorphic functions

Abstract

Let $K$ be a compact subset of the complex $n$-space and $A(K)$ the algebra of all continuous functions on $K$ which are holomorphic on the interior of $K$. In this paper we show that under some hypotheses on $K$, there exists no linear isometry of finite codimension on $A(K)$. Several compact subsets including the closure of strictly pseudoconvex domain and the product of the closure of plane domains which are bounded by a finite number of disjoint smooth curves satisfy the hypotheses.