Vol. 49, No. 2, 1973

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Almost isometries of Banach spaces and moduli of planar domains

Richard Rochberg

Vol. 49 (1973), No. 2, 445–466

Let C(X) and C(Y ) be the supremum normed Banach spaces of continuous complex valued functions on the compact Hausdorff spaces X and Y respectively. Let A and B be closed subspaces of C(X) and C(Y ) respectively. A map from A to B will mean a continuous invertible linear map of A to B. The set of all such maps will be denoted by L(A,B). For T in L(A,B) define c(T) = 1(T∥∥T1). A generalization of the Banach-Stone theorem is proved which shows that there is a constant d < 1 such that if A and B satisfy certain additional technical restrictions and there is a T in L(A,B) with c(T) > d then X and Y are homeomorphic. Furthermore, T is, roughly, composition with this homeomorphism.

For S a connected subset of C bounded by a finite number of disjoint Jordan curves, denote by A(S) the Banach space of functions in C(S) which are analytic on the interior of S. For two such domains, S and S, set d(S,S) = inf {− log c(T );T a linear map of A(S) onto A (S′)}. By analyzing maps T for which c(T) is nearly one, it is shown that d(,) is a metric on the space of moduli of such domain (considered as Riemann surfaces) and that this metric induces the classical moduli topology.

Mathematical Subject Classification 2000
Primary: 30A98
Secondary: 46E15
Received: 31 January 1972
Revised: 25 February 1973
Published: 1 December 1973
Richard Rochberg
Department of Mathematics
Washington University in St. Louis
Campus Box 1146
One Brookings Dr
Saint Louis MO 63130-4899
United States