Some Banach spaces on which all biholomorphic automorphisms are linear

Abstract

The question of whether biholomorphic maps are linear has been treated in various forms by several authors. In particular, Harris [I ] has shown that a biholomorphic map of the unit ball of one space to another which takes 0 to 0 is a restriction of a linear isometry between the two spaces. He then showed that if the unit ball of a Banach space is a homogeneous domain, then it is holomorphically equivalent to the unit ball of another Banach space if and only if the two spaces are isometrically isomorphic. He asked whether this result would hold without the assumption about a homogeneous domain. Kaup and Upmeier [2] gave an answer to this question by showing that two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent. In a recent paper. Stacho [5 1 gave a short proof of the fact that all biholomorphic automorphisms of the unit ball in certain Lp-spaces are linear. In the present note, we show how Stacho’s method can be used to obtain the same result for the C,-spaces and the spaces LP(R. E), where E is an arbitrary Banach space. R is a-finite, and 1 <p < + co, p # 2. In particular. for the discrete case, we get the result that I,(E) has the linear biholomorphic property whether E has the property or not. On the other hand, we show that c,(E) has the property if and only if E has it. A function Q on the open unit ball B(E) of a Banach space is said to be holomorphic in B(E) if the Frechet derivative D@(x, .) of 4 at x exists as a bounded linear map of E into E for each x E B(E). A function 4 from B(E) to B(E) is biholomorphic if 4-l exists and both + and # ’ are holomorphic. The proofs of Theorems 1 and 2 below as well as the theorem in IS] are based on the following lemma proved by Stacho in 1-S 1.