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Zbl 1193.46006
Hatori, Osamu; Kasuga, Kazuhiro
Linear isometries of finite codimensions on Banach algebras of holomorphic functions. (English)
[J] Banach J. Math. Anal. 3, No. 2, 109-124, electronic only (2009). ISSN 1735-8787/e

Summary: 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.

MSC 2000:: *46B04 Isometric theory of Banach spaces
32A38 Algebras of holomorphic functions of several variables
46J15 Banach algebras of differentiable functions

Keywords: shift operators; isometries; uniform algebra

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